Write the Equation from a Graph

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Question 1 of 2

1. Question
Find the equation of the function below by using the graph for $y = x^{3}$ $y=x^3$ .
- 1. $y = {(x + 3)}^{3} - 1$ $y=(x+3)^3 -1$
- 2. $y = {(x - 3)}^{3} + 1$ $y=(x-3)^3 +1$
- 3. $y = {(x - 3)}^{3} - 1$ $y=(x-3)^3 -1$
- 4. $y = {(x + 3)}^{3} + 1$ $y=(x+3)^3 +1$
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Horizontal and vertical translations of cubic functions are written in the form $y = {(x - h)}^{3} + c$ $y=(x-color(red)(h))^3 + color(blue)(c)$ where the point $(h, c)$ $(color(red)(h),color(blue)(c))$ is the vertex of the function.

$- h$ $color(red)(-h)$ is a shift to the right and $+ c$ $color(blue)(+c)$ is a shift upwards.

$(-h) \ bb(rarr)$ Shift Right

$(+h) \ bb(larr)$ Shift Left

$(-c) \ bb(darr)$ Shift Down

$(+c) \ bb(uarr)$ Shift Up

To obtain the equation of the function by using the graph for $y=x^3$ , first sketch the function $y=x^3$ .

Sketch the function $y=x^3$ . Remember the formula $y=(x-color(red)(h))^3 + color(blue)(c)$ when applied to $y=x^3$ (can be rewritten as $y=(x-color(red)(0))^3+color(blue)(0)$ ) has its vertex at $(color(red)(0),color(blue)(0))$ .

To find the horizontal shift ( $h$ ), count the units between the graphs along the $x$ -axis. It is $3$ units to the right ( $h=3$ ).

To find the vertical shift ( $c$ ), count the units between the graphs along the $y$ -axis. It is $1$ unit down ( $c=-1$ ).

Put the equation together using the formula $y=(x-color(red)(h))^3 + color(blue)(c)$ , $color(red)(h=3)$ , and $color(blue)(c=-1)$ . The unknown graph is $y=(x-color(red)(3))^3 color(blue)(-1)$ .

$y=(x-3)^3 -1$
Question 2 of 2

2. Question
Find the equation of the function below by using the graph for $y=1/x$ .
- 1. $y=1/(x-3)+2$
- 2. $y=1/(x+2)-3$
- 3. $y=1/(x-2)+3$
- 4. $y=1/(x+3)-2$
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Horizontal and vertical translations of hyperbolic functions are written in the form $y=1/(x-color(red)(h)) +color(blue)(c)$ where the point $(color(red)(h),color(blue)(c))$ is the intersection point of the asymptotes of the function.

$color(red)(-h)$ is a shift to the right and $color(blue)(+c)$ is a shift upwards.

$(-h) \ bb(rarr)$ Shift Right

$(+h) \ bb(larr)$ Shift Left

$(-c) \ bb(darr)$ Shift Down

$(+c) \ bb(uarr)$ Shift Up

To obtain the equation of the function by using the graph for $y=1/x$ , first sketch the function $y=1/x$ .

Sketch the function $y=1/x$ . Remember the formula $y=1/(x-color(red)(h)) +color(blue)(c)$ when applied to $y=1/x$ (can be rewritten as $y=1/(x-color(red)(0))+color(blue)(0)$ ) has the intersection point of the asymptotes at $(color(red)(0),color(blue)(0))$ .

To find the horizontal shift ( $h$ ), count the units between the graphs along the $x$ -axis. It is $2$ units to the right ( $h=2$ ).

To find the vertical shift ( $c$ ), count the units between the graphs along the $y$ -axis. It is $3$ units up ( $c=3$ ).

Put the equation together using the formula $y=1/(x-color(red)(h)) +color(blue)(c)$ , $color(red)(h=2)$ , and $color(blue)(c=3)$ . The unknown graph is $y=1/(x-color(red)(2))+color(blue)(3)$ .

$y=1/(x-2)+3$

Write the Equation from a Graph

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