Question

F has a range of (-infinaty, 7),

and a graph that is symmetric
about the line x = 3, and has a
Y-intercept of 4

Answer 1

`f(x)=-|x-3|+7`

Explanation:

The parent function of an absolute function is the modulus function:

`f(x)=|x|`

Graph of Modulus function:

(see 2nd attachment)

`f(x)=|x|=\begin{cases} \textsf{\:\:\:$x$, \quad if $x\geq 0$}\\\textsf{$-x$, \quad if $ x < 0$} \end{cases}`

Its vertex is at the origin (0, 0).

Its line of symmetry is x = 0.

It opens upwards.

Given information:

• Range: (-∞, 7)
• Axis of symmetry: x = 3
• y-intercept: (0, 4)

The axis of symmetry is the x-value of the vertex of an absolute function.

The y-value of the range is the y-value of the vertex of an absolute function.

Therefore, the vertex of the absolute function f(x) is (3, 7).

Graphing

• Plot the vertex at (3, 7) and the y-intercept at (0, 4).
• As the function f(x) has a range of (-∞, 7), it opens downwards.
• Draw a straight line through the y-intercept that terminates at the vertex.
• Draw a line that is symmetrical to this line.

To find the equation of the absolute function, work out the series of transformations that take the parent function to the given function.

As the function f(x) has a range of (-∞, 7), it opens downwards. Therefore, the parent function has been reflected in the x-axis:

`\implies -f(x)=-|x|`

As the function f(x) is symmetric about the line x=3, the parent function has been translated 3 units to the right:

`\implies -f(x-3)=-|x-3|`

Finally, as the highest y-value of the function is y = 7, the y-value of its vertex is y = 7. Therefore, the parent function has been translated 7 units up:

`\implies -f(x-3)+7=-|x-3|+7`

Therefore, the equation of the absolute function with the given parameters is:

`f(x)=-|x-3|+7`