4.06 Modulus function
Consider the function y = \left\vert x \right\vert.
What values of x can be substituted into the given function?
What values of y could the function have?
Consider the function y = f\left(x\right) that has been graphed.
Find f\left(2\right).
Find f\left(-2\right).
Write a rule for the function in the form y = \left\vert ax \right\vert.
Write a rule for the function in the form y = a\left\vert x \right\vert.
Consider the function y = \left\vert x - 3 \right\vert.
What is the lowest possible value that this function can have?
What is the highest possible value that this function can have?
Consider the graph of the function f \left( x \right).
State the coordinates of the vertex.
State the equation of the line of symmetry.
Find the gradient of the function for
x \gt 0.
Find the gradient of the function for
x \lt 0.
Is the graph of f\left(x\right) as steep as the graph of y = \left\vert x \right\vert?
Consider the graph of the function f \left( x \right).
State the coordinate of the vertex.
State the equation of the line of symmetry.
Find the gradient of the function for
x \gt 2.
Find the gradient of the function for
x \lt 2.
Is the graph of f \left( x \right) more or less steep than the graph of y = \left\vert x \right\vert?
Consider the function y = \left\vert x \right\vert - 5.
Does the graph of the function open upwards or downwards?
State the coordinate of the vertex.
State the equation of the line of symmetry.
State whether the following functions have narrower graphs than y = \left\vert x \right\vert - 5.
y = \dfrac{\left\vert x \right\vert}{2} - 5
y = \left\vert x - 5 \right\vert
y = \left\vert x \right\vert - 5
y = - 4 \left\vert x \right\vert - 5
Is the function f \left( x \right) = \left\vert 2 x - 2 \right\vert one-to-one?
Consider the following graphs of y = -3x-6 and y = 3x+6:
Rewrite the function f \left( x \right) = \left\vert 3 x + 6 \right\vert as a piecewise function of the form:
f\left(x\right) = \begin{cases} ⬚ & \text{when}\ x\lt ⬚ \\ ⬚ & \text{when}\ x\geq ⬚ \\ \end{cases}What is the vertex of f \left( x \right)?
Consider the function y = \left\vert x+1 \right\vert.
Complete the given table.
Hence graph the function.
State the equation of the axis of symmetry.
State the coordinates of the vertex.
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Complete the given table by writing the equation and gradient for the two lines that make up the graph of the function.
| Equation | Gradient | |
|---|---|---|
| x \lt -1 | ||
| x \gt -1 |
For each of the following functions:
Does the graph of the function open upwards or downwards?
State the coordinates of the vertex.
Sketch the graph.
y = - \left\vert x \right\vert
y = - \left\vert \dfrac{x}{7} \right\vert
Consider the function y = \left\vert 6 x \right\vert.
Does the graph of the function open upwards or downwards?
State the equation of the line of symmetry.
State the coordinates of the vertex.
Graph the function.
State whether the following modulus functions would have a narrower graph than
y = \left\vert 6 x \right\vert:
y = - \left\vert 6 x \right\vert
y = \left\vert 4 x \right\vert
y = \left\vert 7 x \right\vert
y = \left\vert 9 x \right\vert + 3
Graph the following functions:
y = \left\vert 5 x \right\vert
y = \left\vert x - 5 \right\vert
y = - \left\vert 3 x - 6 \right\vert
y = \left\vert 5 - x \right\vert
y = \left\vert \dfrac{x}{3} - 2 \right\vert
y = 8 - \left\vert 4 x \right\vert
Consider the function f \left( x \right) = 11 - \left\vert x \right\vert.
Graph the function.
What is the maximum value of f \left( x \right)?
On which interval is the function increasing?
On which interval is the function decreasing?
Consider the function y = 3 \left\vert x \right\vert - 3.
Complete the table.
Hence graph the function.
State the equation of the axis of symmetry.
What are the coordinates of the vertex?
What are the x-intercepts?
What is the y-intercept?
What is the gradient of the line to the left of the vertex?
What is the gradient of the line to the right of the vertex?
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Consider the function y = \left\vert x + 2 \right\vert + 5.
Graph the function.
What are the coordinates of the vertex?
What is the minimum value of the function?
An absolute value function has its vertex at \left( - 1 , 3\right), and goes through the point A \left(2, 0\right).
Graph the function.
Find the gradient between the vertex and point A.
Hence state the equation of the function.
A modulus function f \left( x \right) has its vertex at \left( - 4 , 0\right), and goes through the point A \left(1, - 5 \right).
Find the gradient between the vertex and point A.
Hence state the equation of the function f \left( x \right).
Michael is training for the land speed record and takes his new car for a test drive by driving straight down a closed highway and back. His distance y in kilometres from the end of the highway x minutes after he takes off is given by the function y = \left\vert 5 x - 40 \right\vert which is graphed below.
How far does he drive in total?
How long does Michael take to reach the end of the highway?
The next day Michael goes for another drive down the same route and his distance from the end of highway is given by
y = \left\vert 4 x - 40 \right\vert which is graphed.
Is Michael driving faster or slower than the previous day?