Graphs of Absolute Value Functions
We will be graphing absolute value functions of the form \(f\left(x\right)=a\left|x-h\right|+k\).
The parent function \(f\left(x\right)=\left|x\right|\) is graphed in red. Using the sliders, explore the graph of \(f\left(x\right)=a\left|x-h\right|+k\) and determine the effect of \(a\), \(h\) and \(k\) on the parent function.
The parent function \(f\left(x\right)=\left|x\right|\) is graphed in red. Using the sliders, explore the graph of \(f\left(x\right)=a\left|x-h\right|+k\) and determine the effect of \(a\), \(h\) and \(k\) on the parent function.
Quick Check 1) How does the graph change if \(a\) is negative? 2) How does the graph change if \(\left|a\right|>1\)? 3) How does the graph change if \(0<\left|a\right|<1\)? 4) How does \(h\) change the graph? 5) How does \(k\) change the graph? Quick Check Solutions |
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Properties of the absolute value function \(f\left(x\right)=a\left|x-h\right|+k\)
*Note, the parentheses are used to indicate the value is not part of the domain and the square brackets are used to indicate the value is included in the domain.
- The vertex of the graph is the point \(\left(h,k\right)\). This is the point where the graph changes direction.
- The axis of symmetry is the line \(x = h\). This is the line, through the vertex, that divides the graph into two identical halves.
- The domain (input) of the function is \(\left\{x\ \mid x\in R\right\}\). The domain represents all values that can be substituted for \(x\). The notation above indicates that \(x\) is the set of all real numbers. Another way this can be represented is with the notation \(\left(-\infty,\infty\right)^*\)
- The range output) of the function depends on the y-coordinate of the vertex and the direction that the function opens. If the graph “opens up”, the range is \(\left\{y\mid y\ge k\right\}\), which can also be represented as \(\left[k,\infty\right)^*\). If the graph “opens down”, the range is \(\left\{y\mid y\le k\right\}\), which can also be represented \(as \left(-\infty,k\right]^*\).
*Note, the parentheses are used to indicate the value is not part of the domain and the square brackets are used to indicate the value is included in the domain.
To graph the absolute value function, plot the vertex. You are graphing two rays whose initial point is the vertex. From this point, you will use \(\large\frac{rise}{run}\) to find other points. You will need to run in both a positive direction (right) and a negative direction (left). Whether you rise up or down depends on the direction the absolute value function opens, so pay attention to the sign of the \(a\) value.
Example 1: Graph the absolute value function \(f\left(x\right)=-\frac{2}{3}\left|x-4\right|+5\). Describe the transformations from the parent function \(f\left(x\right)=\left|x\right|\). State the vertex, axis of symmetry, domain and range of the translated function.
Example 1: Graph the absolute value function \(f\left(x\right)=-\frac{2}{3}\left|x-4\right|+5\). Describe the transformations from the parent function \(f\left(x\right)=\left|x\right|\). State the vertex, axis of symmetry, domain and range of the translated function.
Example 2: Write the equation of the absolute value function graphed below.
\(f\left(x\right)=2\left|x-\left(-1\right)\right|+2\) \(f\left(x\right)=2\left|x+1\right|+2\) |