Question

graph the equation.
$y = 5|x| + 2$

Explanation:

Step1: Identify the vertex

The general form of an absolute - value function is \(y = a|x - h|+k\), where \((h,k)\) is the vertex. For the function \(y = 5|x|+2\), \(h = 0\) and \(k = 2\). So the vertex is at the point \((0,2)\). We can plot this point on the coordinate plane.

Step2: Find other points for \(x\geq0\)

When \(x\geq0\), the absolute - value function \(y = 5|x|+2\) simplifies to \(y = 5x+2\) (since \(|x|=x\) for \(x\geq0\)).

• Let \(x = 0\): \(y=5(0)+2 = 2\) (we already have the vertex).
• Let \(x = 1\): \(y=5(1)+2=5 + 2=7\). So we have the point \((1,7)\).
• Let \(x = 2\): \(y=5(2)+2 = 10 + 2=12\) (but our graph's \(y\) - axis only goes up to 10, so we can also use \(x = 0.5\): \(y=5(0.5)+2=2.5 + 2 = 4.5\), point \((0.5,4.5)\))

Step3: Use symmetry for \(x<0\)

The absolute - value function \(y = a|x|+k\) is symmetric about the \(y\) - axis. So for every point \((x,y)\) with \(x>0\), the point \((-x,y)\) will also be on the graph.

• For \(x = 1\), the point is \((1,7)\), so the point \((- 1,7)\) is also on the graph.
• For \(x = 0.5\), the point is \((0.5,4.5)\), so the point \((-0.5,4.5)\) is also on the graph.

Step4: Draw the graph

We first plot the vertex \((0,2)\). Then we plot the points we found for \(x\geq0\) (e.g., \((1,7)\), \((0.5,4.5)\)) and their symmetric counterparts for \(x<0\) (e.g., \((-1,7)\), \((-0.5,4.5)\)). Then we draw two rays: one for \(x\geq0\) passing through the points with \(x\geq0\) and one for \(x<0\) passing through the points with \(x<0\), both starting from the vertex \((0,2)\). The graph should be a V - shaped graph (since \(a = 5>0\), the graph opens upwards) with the vertex at \((0,2)\), steeper than the parent function \(y=|x|\) (because \(|a| = 5>1\)) and shifted up by 2 units.

Answer:

To graph \(y = 5|x|+2\):

1. Plot the vertex \((0,2)\).
2. For \(x\geq0\), use \(y = 5x + 2\) to find additional points (e.g., \((1,7)\), \((0.5,4.5)\)) and plot them.
3. Use the symmetry about the \(y\) - axis to plot the corresponding points for \(x<0\) (e.g., \((-1,7)\), \((-0.5,4.5)\)).
4. Draw two rays from the vertex through the plotted points, forming a V - shaped graph opening upwards.