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Algebra 1 : How to graph an absolute value function

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Example Question #1 : How To Graph An Absolute Value Function

Absolute_value

Which of these would most likely be the equation corresponding to the above graph?

Possible Answers:

$\displaystyle y = |x - 2 | -1$

$\displaystyle y = |x - 2 | +1$

$\displaystyle y = |x + 2 | -1$

$\displaystyle y = |x + 3 |$

$\displaystyle y = |x + 2 | +1$

Correct answer:

$\displaystyle y = |x - 2 | -1$

Explanation:

This is an absolute value graph. Its equation takes the form $\displaystyle y = |x-h|+k$, in which h,k $\displaystyle h,k$ represent the number of units that the base graph y = |x| $\displaystyle y = |x|$ is translated right and up respectively.

Since the graph of y = |x| $\displaystyle y = |x|$ is translated two units right and one unit down, h = 2 $\displaystyle h = 2$ and k = -1 $\displaystyle k = -1$, so the equation would be:

$\displaystyle y = |x-2|+(-1)$

$\displaystyle y = |x - 2 | -1$

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Example Question #42 : Functions And Lines

Give the $\displaystyle x$-intercept(s) of the graph of the function $f(x) = 7 - \left | 2x-3 \right |$ $\displaystyle f(x) = 7 - \left | 2x-3 \right |$

Possible Answers:

The graph has no $\displaystyle x$-intercepts.

$\displaystyle (-5,0) , (2,0)$

$\displaystyle (-2,0) , (2,0)$

$\displaystyle (-2,0) , (5,0)$

$\displaystyle (-5,0) , (5,0)$

Correct answer:

$\displaystyle (-2,0) , (5,0)$

Explanation:

To find the $\displaystyle x$-intercept(s) of the graph, set $\displaystyle f(x) = 0$ and solve for $\displaystyle x$.

$f(x) = 7 - \left | 2x-3 \right |$ $\displaystyle f(x) = 7 - \left | 2x-3 \right |$

$0= 7 - \left | 2x-3 \right |$ $\displaystyle 0= 7 - \left | 2x-3 \right |$

$\left | 2x-3 \right |= 7$ $\displaystyle \left | 2x-3 \right |= 7$

Rewrite this as the compound equation:

$\displaystyle 2x-3= -7$ or 2x-3= 7 $\displaystyle 2x-3= 7$

Solve each separately:

$\displaystyle 2x-3= -7$

$\displaystyle 2x-3 + 3 = -7+ 3$

2x = -4 $\displaystyle 2x = -4$

$2x \div 2 = -4\div 2$ $\displaystyle 2x \div 2 = -4\div 2$

x = -2 $\displaystyle x = -2$

2x-3= 7 $\displaystyle 2x-3= 7$

$\displaystyle 2x-3 + 3 = 7+ 3$

2x = 10 $\displaystyle 2x = 10$

$2x \div 2 = 10\div 2$ $\displaystyle 2x \div 2 = 10\div 2$

x = 5 $\displaystyle x = 5$

There are two $\displaystyle x$-intercepts: $\displaystyle (-2,0) , (5,0)$

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Example Question #43 : Graphing

Absolute_value

Which of these would most likely be the equation corresponding to the above graph?

Possible Answers:

$\displaystyle y = | x + 6 | - 3$

$\displaystyle y = | x - 6 | - 3$

$\displaystyle y = | x + 3 | - 6$

$\displaystyle y = | x - 3 | - 6$

$\displaystyle y = | x - 6 | + 3$

Correct answer:

$\displaystyle y = | x + 3 | - 6$

Explanation:

Since the graph of y = |x| $\displaystyle y = |x|$ is translated three units left and six units down, h = -3 $\displaystyle h = -3$ and k = -6 $\displaystyle k = -6$.

Plug these values into the general form of the equation:

$\displaystyle y = |x-(-3)|+(-6)$

Simplify:

$\displaystyle y = |x+3|-6$

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Example Question #1 : How To Graph An Absolute Value Function

$\textup{What is the slope of } y=-4\left | x-2\right | \textup{ when } x=1?$ $\displaystyle \textup{What is the slope of } y=-4\left | x-2\right | \textup{ when } x=1?$

Possible Answers:

$\displaystyle -4$

$\displaystyle 4$

$\displaystyle -1$

$\displaystyle 1$

$\textup{It does not exist.}$ $\displaystyle \textup{It does not exist.}$

Correct answer:

$\displaystyle 4$

Explanation:

$\textup{When }x=1\textup{ the expression inside the absolute value is negative,} \\\textup{so the function can be written as } y=-4(2-x)=4x-8,\textup{ with a slope of 4.}\\\textup{Plus, the slope changes to -4 when } x>2. \textup{ When } x=2 \textup{ it does not exist.}$ $\displaystyle \textup{When }x=1\textup{ the expression inside the absolute value is negative,} \\\textup{so the function can be written as } y=-4(2-x)=4x-8,\textup{ with a slope of 4.}\\\textup{Plus, the slope changes to -4 when } x>2. \textup{ When } x=2 \textup{ it does not exist.}$

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Example Question #1 : How To Graph An Absolute Value Function

$\textup{Find the lowest attainable value of }y \textup{ in } y=-4+ \left |4x^2-25 \right |.$ $\displaystyle \textup{Find the lowest attainable value of }y \textup{ in } y=-4+ \left |4x^2-25 \right |.$

Possible Answers:

-2.5 $\displaystyle -2.5$

$\displaystyle -4$

$\displaystyle 2$

$\displaystyle 0$

2.5 $\displaystyle 2.5$

Correct answer:

$\displaystyle -4$

Explanation:

$\\ \left | 4x^2-25\right | \textup{cannot get any lower than zero,}\\\textup{so\textit{ y} cannot get any lower than} -4.$ $\displaystyle \\ \left | 4x^2-25\right | \textup{cannot get any lower than zero,}\\\textup{so\textit{ y} cannot get any lower than} -4.$

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Example Question #46 : Functions And Lines

$\textup{Which is an \textit{x}-intercept of the function } y=\left | -x^2-8x-16\right |-4.$ $\displaystyle \textup{Which is an \textit{x}-intercept of the function } y=\left | -x^2-8x-16\right |-4.$

Possible Answers:

$\displaystyle -2$

$\displaystyle -4$

$\displaystyle 0$

$\textup{None of the above.}$ $\displaystyle \textup{None of the above.}$

$\displaystyle 4$

Correct answer:

$\displaystyle -2$

Explanation:

$\\ \textup{The function can be simplified as } \\y=\left | -(x+4)^2\right |-4=\left |(x+4)^2\right |-4=(x+4)^2-4.\\\textup{In order to find the \textit{x}-intercecpt, we solve the equation}\\0=(x+4)^2-4 \textup{ which becomes }\$x+4)=\pm2\textup{ and has solution } -2 \textup{ and } -6.$ \(\displaystyle \\ \textup{The function can be simplified as } \\y=\left | -(x+4)^2\right |-4=\left |(x+4)^2\right |-4=(x+4)^2-4.\\\textup{In order to find the \textit{x}-intercecpt, we solve the equation}\\0=(x+4)^2-4 \textup{ which becomes }\\(x+4)=\pm2\textup{ and has solution } -2 \textup{ and } -6.$

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Algebra 1 : How to graph an absolute value function

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : How To Graph An Absolute Value Function

Example Question #42 : Functions And Lines

Example Question #43 : Graphing

Example Question #1 : How To Graph An Absolute Value Function

Example Question #1 : How To Graph An Absolute Value Function

Example Question #46 : Functions And Lines

All Algebra 1 Resources

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