Graphing Absolute Value Equations

Graphing Absolute Value Equations exercise

• Summarize the characteristics of the absolute value graph.

  Hints

  Examine the graph above.

  Starting at the origin, the graph consists of two lines:

  • one is given by $y=x$ on the right
  • the other is given by $y=-x$ on the left

  If you flip the right part of the graph, you get the left part.

  Solution

  How do you graph an absolute value equation?

  Let's start with $y=|x|$. This is the parent function for all absolute value functions.

  For $x=1$, $x=2$, $x=3$, the $y$-values are the same as for a linear equation. On the left side of the y-axis, however, for $x$-values that are negative like $x=-1$, $x=-2$, and $x=-3$, the absolute value is positive for the $y$-values such as $y=1$, $y=2$, $y=3$, and so on. The graph looks like a v, and all absolute value function graphs have a v-shape.

  For absolute value graphs, the vertex is the lowest or the highest part of the graph, and the axis of symmetry is the invisible vertical line that passes through the vertex.

• Explain how to determine the equation of the given absolute value function.

  Hints

  The graph of $y=|x+2|$ is two units to the left compared to the graph of $y=|x|$.

  Think about the differences between adding or subtracting a value inside or outside the absolute value bars.

  $|x|+2$ means moving up two units compared to the parent function.

  Solution

  We can recognize absolute value graphs because of their v-shaped forms so we know that this is the graph of an absolute value function.

  Let's start with the parent function, $y=|x|$.

  As the vertex is $(6,0)$, we know to subtract $6$ inside the absolute value bars. Outside, we don't have to add or subtract any other value:

  • $y=|x-6|$

  Let's continue by checking the rise and run of the given line:

  • $m=\frac42=2$

  So we have to multiply the function by $2$, to get the final equation that represents the graph as shown:

  • $y=2|x-6|$

• Identify the differences between the new equations and the parent function, $y=|x|$.

  Hints

  • adding and subtracting inside the absolute value bars $\rightarrow$ shift left or right along the x-axis
  • adding and subtracting outside the absolute value bars $\rightarrow$ shift up or down along the y-axis

  Shifting up or down changes the y-coordinate of the vertex.

  Solution

  By starting with the parent function, $y=|x|$, we are able to determine any graph of an absolute value equation:

  • adding or subtracting inside the absolute value bars results in a shift along the x-axis
  • adding or subtracting outside the absolute value bars results in a shift up (adding) or down (subtracting), along the y-axis
  • multiplying by a factor greater than $1$ results in a narrower graph
  • multiplying by a positive factor less than $1$ results in a wider graph
  • multiplying by a negative factor results in a horizontal flip of the graph

  Compared to $y=|x|$, the graph of $\mathbf{y=|x+5|}$ is shifted five to the left with the new vertex $(-5,0)$.

  The graph of $\mathbf{y=|x|+5}$ is shifted up five with the new vertex $(0,5)$.

  The graph of $\mathbf{y=5|x|}$ is narrower and the vertex stays the same $(0,0)$.

  The graph of $\mathbf{y=|x|-5}$ is shifted down five with the new vertex $(0,-5)$.

• Figure out the equation.

  Hints

  Take a look at the vertex $(1,1)$.

  Compared to the parent function, $y=|x|$, the vertex is shifted

  • one unit to the left
  • one unit up

  Solution

  As usual, let's compare this graph with the parent function, $y=|x|$.

  In our picture with the flashlight, the vertex is given by $(1,1)$.

  1. First, we will take a look at the x-coordinate of the vertex $x=1$. That tells us that we have to shift the function one unit to the right. We can write this as the following: $y=|x+1|$.
  2. Now, we can check the rise and run, or slope: we move up two units (rise) and we move three units to the right (run). So we get the factor $\frac23$. This gives us $y=\frac23|x+1|$.
  3. Finally, we still have to deal with the y-coordinate of the vertex, $y=1$. The graph is shifted up one unit, which means adding $1$ outside the absolute value bars: $y=\frac23|x+1|+1$.

• Select the graph corresponding to $y=|x|+1$.

  Hints

  Adding or subtracting any value inside the absolute value bars will shift the graph left or right along the x-axis.

  Adding or subtracting any value outside the absolute value bars will shift the graph up or down along the y-axis.

  Multiplying by a factor greater than $1$ leads to a narrower graph, whereas multiplying by a factor less than $1$ but more than 0 leads to a wider graph.

  Solution

  Starting with the parent function, $y=|x|$,

  • adding or subtracting inside the absolute value bars results in a shift along the x-axis
  • adding or subtracting outside the absolute value bars results in a shift along the y-axis
  • multiplying or dividing outside the absolute value bars leads to a narrower or wider graph

  $\mathbf{y=|x+1|}$: The graph is the parent function shifted one unit to the left. This is the blue graph.

  $\mathbf{y=|x-2|}$: The graph is the parent function shifted two units to the right. This is the green graph.

  $\mathbf{y=|x|+1}$: The graph is the parent function shifted one unit up. This is the red graph.

  $\mathbf{y=3|x|}$: The graph is narrower than the parent graph. This is the violet graph.

  $\mathbf{y=\frac12|x|}$: The graph is wider than the parent graph. This is the orange graph.

  The turquoise graph is a line corresponding to the linear equation $y=3x+2$.

• Label the absolute value equations.

  Hints

  Start by determining the vertex of each graph in relation to the parent function, $y=|x|$.

  Once you know the vertex, you know how each graph shifts:

  • along the x-axis - to the left or to the right
  • along thy y-axis - up or down

  $m$ in $y=m|x|$ is determined by rise over run.

  Use the following approach:

  1. determine the x-axis shift
  2. consider the factor if necessary
  3. determine the y-axis shift

  Solution

  Let's determine the equation of each absolute value graph by using the following steps:

  1. check the vertex
  2. determine the x-axis shift depending on the vertex
  3. calculate the slope by counting the rise over run
  4. determine the y-axis shift depending on the vertex

  The purple line

  • the vertex is given by $(-2,1)$
  • x-axis-shift: $y=|x+2|$
  • $m=\frac11=1$
  • y-axis-shift: $\mathbf{y=|x+2|+1}$

  The blue line

  • the vertex is given by $(-3,0)$
  • x-axis shift: $y=|x+3|$
  • $m=\frac31=3$
  • we have no y-axis shift
  • $\mathbf{y=3|x+3|}$

  The green line

  • the vertex is given by $(0,2)$
  • we have no x-axis shift
  • $m=\frac12$
  • y-axis shift: $\mathbf{y=\frac12|x|+2}$

  The orange line

  • The vertex is given by $(2,1)$
  • x-axis shift: $y=|x-2|$
  • $m=\frac21=2$
  • y-axis shift: $\mathbf{y=2|x-2|+1}$