4.07 Absolute value functions
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?
What is the domain of the function f \left( x \right) = \left\vert 6 - x \right\vert?
Consider the function that has been graphed.
What is the domain of the function?
What is the range of the function?
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?
What is the range of the function?
What is the domain of the function?
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 = \left\vert \dfrac{x}{2} - 5 \right\vert
y = \left\vert x - 5 \right\vert
y = \left\vert x \right\vert - 5
y = \left\vert 4x-5 \right\vert
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 domain and range of f \left( x \right)? Give your answers in interval notation.
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 absolute value functions:
Does the graph of the function open upwards or downwards?
State the coordinates of the vertex.
Sketch the graph.
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.
Graph the following absolute value functions:
y = \left\vert 5 x \right\vert
y = \left\vert x - 5 \right\vert
y = \left\vert 5 - x \right\vert
y = \left\vert \dfrac{x}{3} - 2 \right\vert
Consider the quadratic function h \left( x \right) = x^{2} + 2.
Sketch the graph of the parabola y = h \left( x \right).
Sketch the graph of the parabola y = \vert h \left( x \right) \vert.
Explain why there is no difference between the graphs of the functions y=h(x) and y=\vert h(x) \vert.
Consider the parabola y = \left(2 - x\right) \left(x + 4\right).
State the y-intercept.
State the x-intercepts.
Determine the coordinates of the vertex of the parabola.
Sketch the graph of the parabola.
Sketch the function y=\vert (2-x)(x+4) \vert
Consider the parabola y = \left(x - 3\right) \left(x - 1\right).
Find the y-intercept.
Find the x-intercepts.
State the equation of the axis of symmetry.
Find the coordinates of the turning point.
Sketch the graph of the parabola.
Sketch the graph of the function y=|(x-3)(x-1)|.
Consider the parabola y = x \left(x + 6\right).
Find the y-intercept.
Find the x-intercepts.
State the equation of the axis of symmetry.
Find the coordinates of the turning point.
Sketch the graph of the parabola.
Sketch the graph of the function y = \vert x(x+6) \vert.
Sketch the graph of the following:
Consider the function y = \left(x + 5\right) \left(x + 1\right).
Sketch the graph.
Sketch the graph of y = \vert \left(x + 5\right) \left(x + 1\right) \vert on the same set of axes.
Consider the equation y = \vert x^{2} - 6 x + 8 \vert.
Factorise the expression x^{2} - 6 x + 8.
Hence, or otherwise, find the x-intercepts of the quadratic function y = x^{2} - 6 x + 8
Find the coordinates of the turning point.
Sketch the graph of the function y = x^{2} - 6 x + 8.
Sketch the graph of the function y = \vert x^{2} - 6 x + 8 \vert.
Consider the function y = \vert x^{2} + x - 12 \vert.
Find the x-intercepts of the curve.
Find the y-intercept of the curve.
What is the equation of the vertical axis of symmetry for the parabola?
Find the coordinates of the vertex of the parabola.
Sketch the graph of y = x^{2} + x - 12.
Sketch the graph of y = \vert x^{2} + x - 12 \vert.
Consider the function y = \vert x^{2} + 4 x-1 \vert.
Find the y-intercept of the parabola.
Find the vertex of the parabola.
Sketch the graph of y = x^{2} + 4 x-1.
Sketch the graph of y = \vert x^{2} + 4 x-1 \vert.
Sketch the graphs of the following functions: