Graphical Representation of Functions - SAT Math
Card 0 of 224
Which of the following represents a quadratic equation with its zeros at 
and 
?
Which of the following represents a quadratic equation with its zeros at 
The important first step of creating a quadratic equation from its zeros is knowing what a zero really is. A zero of a quadratic (or polynomial) is an x-coordinate at which the y-coordinate is equal to 0.
We find that algebraically by factoring quadratics into the form 
, and then setting 
equal to 
and 
, because in each of those cases and entire parenthetical term would equal 0, and anything times 0 equals 0.
Here the question gives you a head start: we know that the numbers 4 and 5 can go in the 
and 
spots, because if so we'll have found our zeros. So we can set up the equation:
This satisfies the requirements of zeros, but now we need to expand this equation using FOIL to turn it into a proper quadratic. That means that our quadratic is:
And when we combine like terms it's:
The important first step of creating a quadratic equation from its zeros is knowing what a zero really is. A zero of a quadratic (or polynomial) is an x-coordinate at which the y-coordinate is equal to 0.
We find that algebraically by factoring quadratics into the form 
Here the question gives you a head start: we know that the numbers 4 and 5 can go in the 
This satisfies the requirements of zeros, but now we need to expand this equation using FOIL to turn it into a proper quadratic. That means that our quadratic is:
And when we combine like terms it's:
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How many distinct zeros does the function 
have?
How many distinct zeros does the function 
The zeros of a function are points at which the function crosses the x-axis, or perhaps more simply points at which the function is equal to 0. So to solve for those, set the function equal to zero and then solve it like you would a quadratic. Here that gives you:
You can then factor the common 
term:
And then factor the quadratic within parentheses. Note that this quadratic is one of the common perfect square quadratics:
or 
The solutions to this equation, then, are 
and 
. Note that the question asks you for how many distinct zeros the function has, so you cannot count 
twice. The answer, then, is 2.
The zeros of a function are points at which the function crosses the x-axis, or perhaps more simply points at which the function is equal to 0. So to solve for those, set the function equal to zero and then solve it like you would a quadratic. Here that gives you:
You can then factor the common 
And then factor the quadratic within parentheses. Note that this quadratic is one of the common perfect square quadratics:
The solutions to this equation, then, are 
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Which of the following equations belongs to a polynomial function with zeros at 
and 
?
Which of the following equations belongs to a polynomial function with zeros at 
A function with zeros at 2 and 7 would factor to 
, where it is important to recognize that 
is the coefficient. So while you might be looking to simply expand 
to 
, note that none of the options with a simple 
term (and not 
) directly equal that simple quadratic when set to 0.
However, if you multiply 
by 2, you get 
.
A function with zeros at 2 and 7 would factor to 
However, if you multiply 
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How many distinct zeros does the function 
have?
How many distinct zeros does the function 
When you're graphing a function, the zeros of the function are the points at which the function crosses the x-axis. What that really means is that the value of the function is zero at those points, so to solve for those zeros algebraically you can just set the function equal to zero and solve. Here that would mean:
So factor the common 
term to get:
And then factor the quadratic within parentheses:
You can then see that there are three values of 
that would make this equation true: 
and 
. The answer is therefore 3.
When you're graphing a function, the zeros of the function are the points at which the function crosses the x-axis. What that really means is that the value of the function is zero at those points, so to solve for those zeros algebraically you can just set the function equal to zero and solve. Here that would mean:
So factor the common 
And then factor the quadratic within parentheses:
You can then see that there are three values of 
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What is the sum of all unique zeros to the function 
?
What is the sum of all unique zeros to the function 
The zeros of a function are the x-values at which the function is equal to zero. So to solve for the zeros, set the function equal to zero. That would give you here:
Then you can factor the common 
to get:
And then factor like you would a quadratic:
Or, more succinctly formatted:
This means that the zeros are at 
and 
. Now, importantly, look at what the question asks for. It wants the sum (add the zeros) of all UNIQUE zeros, meaning you should not count 
twice. The sum then is 
.
The zeros of a function are the x-values at which the function is equal to zero. So to solve for the zeros, set the function equal to zero. That would give you here:
Then you can factor the common 
And then factor like you would a quadratic:
Or, more succinctly formatted:
This means that the zeros are at 
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The function 
does not have any zeros for its domain of all real numbers. Which of the following must be true?
The function 
The zeros of a function are points at which the function is equal to zero. Here you're told that 
does not equal zero for any real number values of 
and you're asked to determine what that means for 
. Note that since 
cannot be negative, 
would have to be negative in order for the 
term to reduce the other term, 
to zero. Since you know that this function never equals zero, you can conclude that 
is not negative, which means that it is greater than or equal to zero.
The zeros of a function are points at which the function is equal to zero. Here you're told that 
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What is the sum of all zeros to the function 
?
What is the sum of all zeros to the function 
The zeros of a function are the x-values at which the function itself is equal to zero, so you will generally want to solve these problems by setting the function equal to zero and then factoring like a quadratic. Here that means you would start with:
And then factor the common 
term:
You now know that one of the solutions is 
, as that would mean that the entire parenthetical term would be multiplied by zero. But you still need to work within the parentheses to factor that quadratic. You should see some helpful factoring clues: when you factor into two parentheticals, the numeric terms have to multiply to -1, meaning that you will likely have one be 1 and the other -1. And the first terms need to multiply to 
meaning that your first terms will be 
and 
. So you can set up your parentheses as:
, where one 
will have a + and one will have a - sign.If you play with the options to see what will work to give you the 
middle term in 
, you'll see that the proper factorization is:
This means that the solutions for 
are 
and 
. The sum of these solutions, then, is 
.
The zeros of a function are the x-values at which the function itself is equal to zero, so you will generally want to solve these problems by setting the function equal to zero and then factoring like a quadratic. Here that means you would start with:
And then factor the common 
You now know that one of the solutions is 
This means that the solutions for 
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How many distinct zeros does the function 
have?
How many distinct zeros does the function 
The zeros of a function are essentially points (or at least the x-values of the points) at which the function is equal to zero. So to solve for the zeros of a function, first set that function itself equal to zero. Here that would mean:
Then factor like you would a quadratic; since you have it set to zero, if any multiplicative term equals zero then the "equals zero" will hold for the whole equation. First you can factor the common 
term:
And then you can factor the quadratic within:
This then means that the zeros for this function are at 
and 
, meaning that this function has three distinct zeros.
The zeros of a function are essentially points (or at least the x-values of the points) at which the function is equal to zero. So to solve for the zeros of a function, first set that function itself equal to zero. Here that would mean:
Then factor like you would a quadratic; since you have it set to zero, if any multiplicative term equals zero then the "equals zero" will hold for the whole equation. First you can factor the common 
And then you can factor the quadratic within:
This then means that the zeros for this function are at 
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How many unique zeros does the function 
have?
How many unique zeros does the function 
The zeros of a function are the x-coordinates at the points where the function (the y-coordinate, if you're graphing it) equals zero. So to solve for the zeros, set the function equal to zero:
Here you can then factor the common 
term, yielding:
Now you can factor the quadratic within the parentheses. This gives you:
The solutions to this equation are at the points 
and 
, demonstrating that there are 3 unique zeros to this function.
The zeros of a function are the x-coordinates at the points where the function (the y-coordinate, if you're graphing it) equals zero. So to solve for the zeros, set the function equal to zero:
Here you can then factor the common 
Now you can factor the quadratic within the parentheses. This gives you:
The solutions to this equation are at the points 
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What is the sum of all unique zeros to the function 
?
What is the sum of all unique zeros to the function 
The zeros of a function are the x-values at which the value of the function equals zero. So to solve for the zeros of a function, set the function equal to zero and then solve for 
. Here that means setting the function equal to zero:
Then factor the common 
:
Then factor the quadratic in parentheses:
Now make sure you solve for 
. The values of 
that lead to a product of zero are 
and 
. Therefore the sum is 
.
The zeros of a function are the x-values at which the value of the function equals zero. So to solve for the zeros of a function, set the function equal to zero and then solve for 
Then factor the common 
Then factor the quadratic in parentheses:
Now make sure you solve for 
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is a nonzero constant in the quadratic equation above. When graphed in the 
-plane, the equation is a parabola with a vertex at point 
. What is the value of 
?
There are two ways to solve this problem. For one, you might see that the problem involves a vertex and decide to put the equation in vertex form, 
, where the vertex is at point 
. To employ vertex form, recognize that you will need to get the quadratic to be a perfect square, using the 
term to balance. Here if you were to multiply out the quadratic in the given equation, you'd have:
Note that the only perfect square quadratic with 
as the middle term is 
, which factors to the perfect square 
. So in order to put this equation in vertex form, you need to convert 
into 
, where 
equals out the step you took to turn 
within the parentheses to 
. To do that step, you added 
(remember that the entire parenthetical is multiplied by 
). So you need to subtract 
on the outside of the parentheses to keep the equation balanced. That means that you now have:
, or when factored directly to vertex form 
.
This means that the vertex is at 
, or more directly that 
. So the correct answer is 
.
You could also use the line of symmetry to solve here. You know that the zeros of the given equation, 
, will be at 
and 
, as those are the points at which each parenthetical term equals zero, so 
would in turn equal zero. Because parabolas are symmetrical, that tells you that the vertex will lie halfway between those two points, at 
. If you then plug in the 
-coordinate of the vertex, 
, you'll get the corresponding 
-coordinate:
.
There are two ways to solve this problem. For one, you might see that the problem involves a vertex and decide to put the equation in vertex form, 
Note that the only perfect square quadratic with 
This means that the vertex is at 
You could also use the line of symmetry to solve here. You know that the zeros of the given equation, 
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The parabola pictured above is defined by the equation 
and has a vertex at point 
. Which of the following is true of a parabola defined by the equation 
?
The parabola pictured above is defined by the equation 
Vertex form of a parabola is 
, where the point 
is the vertex of the parabola, and the sign of 
determines which way the parabola opens. A negative 
opens downward, and a positive 
opens upward. Here the given parabola has established that the value of 
without the negative sign would point upward, so multiplying 
by 
will flip the parabola to point downward. And in Vertex Form, 
and 
are 
and 
, so the vertex will be at 
.
Vertex form of a parabola is 
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Complete the square to calculate the maximum or minimum point of the given function.
Complete the square to calculate the maximum or minimum point of the given function.
Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,
where when multiplied out,
the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.
Complete the square for this particular function is as follows.
First identify the middle term coefficient.
Now divide the middle term coefficient by two.
From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.
When simplified the new function is,
Since the 
term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the 
value of the vertex set the inside portion of the binomial equal to zero and solve.
From here, substitute the the 
value into the original function.
Therefore the minimum value occurs at the point 
.
Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,
where when multiplied out,
the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.
Complete the square for this particular function is as follows.
First identify the middle term coefficient.
Now divide the middle term coefficient by two.
From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.
When simplified the new function is,
Since the
From here, substitute the the
Therefore the minimum value occurs at the point
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Complete the square to calculate the maximum or minimum point of the given function.
Complete the square to calculate the maximum or minimum point of the given function.
Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,
where when multiplied out,
the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.
Complete the square for this particular function is as follows.
First identify the middle term coefficient.
Now divide the middle term coefficient by two.
From here write the function with the perfect square.
When simplified the new function is,
Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.
From here, substitute the the value into the original function.
Therefore the minimum value occurs at the point 
.
Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,
where when multiplied out,
the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.
Complete the square for this particular function is as follows.
First identify the middle term coefficient.
Now divide the middle term coefficient by two.
From here write the function with the perfect square.
When simplified the new function is,
Since the
From here, substitute the the
Therefore the minimum value occurs at the point
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Complete the square to calculate the maximum or minimum point of the given function.
Complete the square to calculate the maximum or minimum point of the given function.
Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,
where when multiplied out,
the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.
Complete the square for this particular function is as follows.
First factor out a negative one.
Now identify the middle term coefficient.
Now divide the middle term coefficient by two.
From here write the function with the perfect square.
When simplified the new function is,
Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.
From here, substitute the the value into the original function.
Therefore the maximum value occurs at the point 
.
Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,
where when multiplied out,
the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.
Complete the square for this particular function is as follows.
First factor out a negative one.
Now identify the middle term coefficient.
Now divide the middle term coefficient by two.
From here write the function with the perfect square.
When simplified the new function is,
Since the
From here, substitute the the
Therefore the maximum value occurs at the point
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In which quadrant of the xy-plane does the vertex of the following function lie?
In which quadrant of the xy-plane does the vertex of the following function lie?
Finding the vertex of a quadratic function is best accomplished by finding 
and 
in the equation in vertex form where 
, or by leveraging the formulas for 
and 
. So your goal should be to take the given equation and put it in that form. You start with:
Start by factoring out the common 3 to see that:
Recognize that the perfect square for the terms 
will be 
and then adjust the remaining term to keep equivalence:
Lastly, you need to get the – 5 out of the parentheses so that you are in the required form 
. With this last step you see that:
So, 
and 
Because 
and 
, the point 
is 
, meaning that the vertex lies in Quadrant III.
Note: You could also find the vertex by simply leveraging the two formulas below:
Since 
, 
, 
, we can plug into the equation to see that
Thus the vertex is 
.
Finding the vertex of a quadratic function is best accomplished by finding 
Start by factoring out the common 3 to see that:
Recognize that the perfect square for the terms 
Lastly, you need to get the – 5 out of the parentheses so that you are in the required form 
So, 
Because 
Note: You could also find the vertex by simply leveraging the two formulas below:
Since 
Thus the vertex is 
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How many distinct roots does the polynomial 
have?
How many distinct roots does the polynomial 
This equation isn’t at all impossible to factor, but you may not notice that at first glance (or on a future problem you may try to factor and get stuck). If you see this problem on the calculator section, it’s likely most efficient to simply graph it on your calculator and count the number of times that the graph touches the y-axis. Your graph will look like:
As you can see, the graph crosses the y-axis twice and touches it a third time (at the origin, or point (0, 0)), so the answer is 3. Of course, you could factor this equation by factoring out like terms. Factoring out the common x2 term creates: 
, which can then be factored to 
. Since your goal is for y to equal 0 (the definition of “a zero” in a function or a “root” in an equation), you can set that equal to 0: 
. This means that the roots are 0, −6, and 3, for a total of 3 roots.
This equation isn’t at all impossible to factor, but you may not notice that at first glance (or on a future problem you may try to factor and get stuck). If you see this problem on the calculator section, it’s likely most efficient to simply graph it on your calculator and count the number of times that the graph touches the y-axis. Your graph will look like:
As you can see, the graph crosses the y-axis twice and touches it a third time (at the origin, or point (0, 0)), so the answer is 3. Of course, you could factor this equation by factoring out like terms. Factoring out the common x2 term creates: 
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Which of the following represents the maximum or minimum point of the given function?
Which of the following represents the maximum or minimum point of the given function?
Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,
where when multiplied out,
the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.
Complete the square for this particular function is as follows.
First identify the middle term coefficient.
Middle Term Coefficient=2
Now divide the middle term coefficient by two.
From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.
When simplified the new function is,
Since the 
term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the x value of the vertex set the inside portion of the binomial equal to zero and solve.
From here, substitute the the x value into the original function.
Therefore the minimum value occurs at the point (−1,−5).
Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,
where when multiplied out,
the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.
Complete the square for this particular function is as follows.
First identify the middle term coefficient.
Middle Term Coefficient=2
Now divide the middle term coefficient by two.
From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.
When simplified the new function is,
Since the 
From here, substitute the the x value into the original function.
Therefore the minimum value occurs at the point (−1,−5).
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Which of the following is an equivalent form of the function f above for which the minimum value of f appears as a constant or coefficient?
Which of the following is an equivalent form of the function f above for which the minimum value of f appears as a constant or coefficient?
By either graphing the function, utilizing the formula for the vertex, or completing the square, we can see that the minimum value of the function is -49. In order to comply with the parameters of the question stem, this minimum must visually appear in the answer option, so
is our only feasible option. That said, we can prove this process by "completing the square."
Here, if we foil
,
we arrive at
to complete the square, we want to add, and subsequently subtract 4 as shown below
which simplifies to
and can be rewritten as
our equivalent equation in vertex form!
By either graphing the function, utilizing the formula for the vertex, or completing the square, we can see that the minimum value of the function is -49. In order to comply with the parameters of the question stem, this minimum must visually appear in the answer option, so
is our only feasible option. That said, we can prove this process by "completing the square."
Here, if we foil
we arrive at
to complete the square, we want to add, and subsequently subtract 4 as shown below
which simplifies to
and can be rewritten as
our equivalent equation in vertex form!
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Which of the following are distinct zeros does the function 
?
I) -9
II) -4
III) 4
Which of the following are distinct zeros does the function 
I) -9
II) -4
III) 4
To tackle this question, we'll want to see the number sense behind factoring by inspection. In the quadratic construction
we're looking for two numbers that multiply to give us our constant, "c," and sum to give us the coefficient "b."
In this case, those two numbers are 9 and -4, meaning we can reconstruct the function to read
since we're looking for our x intercepts or "zeros," we want the case where our output or "y" = 0, so we can set the righthand side of the function equal to zero as such
in order for this equation to hold true, either of our parentheticals will need to equal zero, meaning that if
or
then x could be equal to -9 or 4. So, I and III are zeros of the function.
*Note - if given this question on a calculator-friendly section, we could graph and identify zeros as well*
To tackle this question, we'll want to see the number sense behind factoring by inspection. In the quadratic construction
we're looking for two numbers that multiply to give us our constant, "c," and sum to give us the coefficient "b."
In this case, those two numbers are 9 and -4, meaning we can reconstruct the function to read
since we're looking for our x intercepts or "zeros," we want the case where our output or "y" = 0, so we can set the righthand side of the function equal to zero as such
in order for this equation to hold true, either of our parentheticals will need to equal zero, meaning that if
or
then x could be equal to -9 or 4. So, I and III are zeros of the function.
*Note - if given this question on a calculator-friendly section, we could graph and identify zeros as well*
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