### All SAT II Math I Resources

## Example Questions

### Example Question #1 : Graphing Linear Functions

Refer to the above red line. A line is drawn perpendicular to that line, and with the same -intercept. Give the equation of that line in slope-intercept form.

**Possible Answers:**

**Correct answer:**

First, we need to find the slope of the above line.

The slope of a line. given two points can be calculated using the slope formula

Set :

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 2, which would be . Since we want this line to have the same -intercept as the first line, which is the point , we can substitute and in the slope-intercept form:

### Example Question #1 : Graphing Functions

Refer to the above diagram. If the red line passes through the point , what is the value of ?

**Possible Answers:**

**Correct answer:**

One way to answer this is to first find the equation of the line.

The slope of a line. given two points can be calculated using the slope formula

Set :

The line has slope 3 and -intercept , so we can substitute in the slope-intercept form:

Now substitute 4 for and for and solve for :

### Example Question #1 : Graphing Linear Functions

Which equation best matches the graph of the line shown above?

**Possible Answers:**

**Correct answer:**

To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.

Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.

Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:

which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:

### Example Question #1 : Graphing Linear Functions

Determine where the graphs of the following equations will intersect.

**Possible Answers:**

**Correct answer:**

We can solve the system of equations using the substitution method.

Solve for in the second equation.

Substitute this value of into the first equation.

Now we can solve for .

Solve for using the first equation with this new value of .

The solution is the ordered pair .

### Example Question #2 : Graphing Linear Functions

Refer to the line in the above diagram. It we were to continue to draw it so that it intersects the -axis, where would its -intercept be?

**Possible Answers:**

**Correct answer:**

First, we need to find the slope of the line.

In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown .

We can use this to find the -intercept using the slope formula as follows:

The lower left point has coordinates . Therefore, we can set up and solve for in this slope formula, setting :

### Example Question #3 : Graphing Linear Functions

Line includes the points and . Line includes the points and . Which of the following statements is true of these lines?

**Possible Answers:**

The lines are distinct but neither parallel nor perpendicular.

The lines are perpendicular.

Insufficient information is given to answer this question.

The lines are identical.

The lines are parallel.

**Correct answer:**

The lines are parallel.

We calculate the slopes of the lines using the slope formula.

The slope of line is

The slope of line is

The lines have the same slope, making them either parallel or identical.

Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form , where is the -coordinate of each point on the line. Therefore, line and line have equations and .This makes them parallel lines.

### Example Question #1 : Graphing Inequalities

Refer to the above diagram. which of the following compound inequality statements has this set of points as its graph?

**Possible Answers:**

**Correct answer:**

A horizontal line has equation for some value of ; since the line goes through a point with -coordinate 3, the line is . Also, since the line is solid and the region above this line is shaded in, the corresponding inequality is .

A vertical line has equation for some value of ; since the line goes through a point with -coordinate 4, the line is . Also, since the line is solid and the region right of this line is shaded in, the corresponding inequality is .

Since only the region belonging to *both* sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is .

### Example Question #1 : Graphing Functions

Which of the following inequalities is graphed above?

**Possible Answers:**

**Correct answer:**

First, we determine the equation of the boundary line. This line includes points and , so the slope can be calculated as follows:

Since we also know the -intercept is , we can substitute in the slope-intercept form to obtain the equation of the boundary line:

The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either or . To find out which one, we can test a point in the solution set - for ease, we will choose :

_____

_____

_____

0 is less than 3 so the correct symbol is .

The inequality is .

### Example Question #1 : Graphing Linear Functions

Select the equation of the line perpendicular to the graph of .

**Possible Answers:**

None of these.

**Correct answer:**

Lines are perpendicular when their slopes are the negative recicprocals of each other such as . To find the slope of our equation we must change it to slope y-intercept form.

Subtract the x variable from both sides:

Divide by 4 to isolate y:

The negative reciprocal of the above slope: . The only equation with this slope is .

### Example Question #2 : Graphing Linear Functions

An individual's maximum heart rate can be found by subtracting his or her age from . Which graph correctly expresses this relationship between years of age and maximum heart rate?

**Possible Answers:**

**Correct answer:**

In form, where y = maximum heart rate and x = age, we can express the relationship as:

We are looking for a graph with a slope of -1 and a y-intercept of 220.

The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.

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