Question

use technology to find points and then graph the function $y = 2|x| + 1$, following the instructions below.

Explanation:

Step1: Analyze the function type

The function \( y = 2|x|+ 1 \) is an absolute - value function. The general form of an absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex of the V - shaped graph. In this function, \( h = 0\) and \( k=1\), so the vertex of the graph is at the point \((0,1)\).

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

When \(x\geq0\), the absolute - value function simplifies to \(y = 2x + 1\) (since \(|x|=x\) for \(x\geq0\)).

• Let \(x = 1\): Substitute \(x = 1\) into \(y=2x + 1\), we get \(y=2\times1 + 1=3\). So the point is \((1,3)\).
• Let \(x = 2\): Substitute \(x = 2\) into \(y = 2x+1\), we get \(y=2\times2 + 1 = 5\). So the point is \((2,5)\).

Step3: Find additional points for \(x<0\)

When \(x<0\), the absolute - value function simplifies to \(y=- 2x + 1\) (since \(|x|=-x\) for \(x < 0\)).

• Let \(x=-1\): Substitute \(x=-1\) into \(y=-2x + 1\), we get \(y=-2\times(-1)+1=2 + 1=3\). So the point is \((-1,3)\).
• Let \(x=-2\): Substitute \(x = - 2\) into \(y=-2x + 1\), we get \(y=-2\times(-2)+1=4 + 1=5\). So the point is \((-2,5)\).

Step4: Graph the function

Plot the vertex \((0,1)\), and the points \((1,3)\), \((2,5)\), \((-1,3)\), \((-2,5)\) on a coordinate plane. Then, draw two straight lines: one line passing through the points \((0,1)\), \((1,3)\), \((2,5)\) (for \(x\geq0\)) and another line passing through the points \((0,1)\), \((-1,3)\), \((-2,5)\) (for \(x<0\)). The graph is a V - shaped graph opening upwards with the vertex at \((0,1)\) and the slope of the right - hand side (for \(x\geq0\)) is \(2\) and the slope of the left - hand side (for \(x < 0\)) is \(- 2\).

Answer:

To graph \(y = 2|x|+1\):

1. Identify the vertex \((0,1)\) (from the form \(y=a|x - h|+k\) with \(h = 0,k = 1,a = 2\)).
2. Find points:

• For \(x\geq0\) (using \(y = 2x + 1\)): \((1,3)\), \((2,5)\)
• For \(x<0\) (using \(y=-2x + 1\)): \((-1,3)\), \((-2,5)\)

1. Plot the vertex and the points, then draw two lines (one for \(x\geq0\), one for \(x < 0\)) to form the V - shaped graph of the absolute - value function.