Question

graph the equation.
$y = 3\vert x - 2\vert + 3$

Explanation:

Step1: Identify the vertex of the absolute - value function

The general form of an absolute - value function is \(y = a|x - h|+k\), where the vertex is \((h,k)\). For the function \(y = 3|x - 2|+3\), by comparing with the general form, we have \(h = 2\) and \(k = 3\). So the vertex of the graph is the point \((2,3)\).

Step2: Determine the slope of the two branches

For the absolute - value function \(y=a|x - h|+k\), when \(x\geq h\), the function can be written as \(y=a(x - h)+k\) (a linear function with slope \(a\)), and when \(x\lt h\), the function can be written as \(y=-a(x - h)+k\) (a linear function with slope \(-a\)). Here, \(a = 3\).

• When \(x\geq2\), \(y = 3(x - 2)+3=3x-6 + 3=3x - 3\). The slope of this line is \(m = 3\).
• When \(x\lt2\), \(y=3(2 - x)+3=6-3x + 3=9 - 3x\). The slope of this line is \(m=- 3\).

Step3: Find two points on each branch

• For the branch with \(x\geq2\) (slope \(m = 3\)):

Let's take \(x = 3\). Substitute \(x = 3\) into \(y = 3x-3\), we get \(y=3\times3 - 3=9 - 3 = 6\). So the point \((3,6)\) is on this branch.
Let's take \(x = 4\). Substitute \(x = 4\) into \(y = 3x-3\), we get \(y=3\times4 - 3=12 - 3 = 9\). So the point \((4,9)\) is on this branch.

• For the branch with \(x\lt2\) (slope \(m=-3\)):

Let's take \(x = 1\). Substitute \(x = 1\) into \(y=9 - 3x\), we get \(y=9-3\times1 = 6\). So the point \((1,6)\) is on this branch.
Let's take \(x = 0\). Substitute \(x = 0\) into \(y=9 - 3x\), we get \(y=9-3\times0 = 9\). So the point \((0,9)\) is on this branch.

Step4: Plot the vertex and the points, then draw the lines

• First, plot the vertex \((2,3)\) on the coordinate plane.
• Then, plot the points \((3,6)\), \((4,9)\) for the right - hand branch (where \(x\geq2\)) and the points \((1,6)\), \((0,9)\) for the left - hand branch (where \(x\lt2\)).
• Draw a line with slope \(3\) passing through the vertex \((2,3)\) and the points \((3,6)\), \((4,9)\) (for \(x\geq2\)) and a line with slope \(- 3\) passing through the vertex \((2,3)\) and the points \((1,6)\), \((0,9)\) (for \(x\lt2\)). The graph will be a V - shaped graph with the vertex at \((2,3)\), opening upwards (since \(a = 3>0\)) and with steeper slopes (because \(|a| = 3>1\)) compared to the parent function \(y = |x|\).

Answer:

To graph \(y = 3|x - 2|+3\):

1. Plot the vertex \((2,3)\).
2. For \(x\geq2\), use the line \(y = 3x - 3\) (plot points like \((3,6)\), \((4,9)\) and draw the line through \((2,3)\) with slope \(3\)).
3. For \(x\lt2\), use the line \(y=9 - 3x\) (plot points like \((1,6)\), \((0,9)\) and draw the line through \((2,3)\) with slope \(-3\)). The resulting graph is a V - shaped graph with vertex at \((2,3)\), opening upwards, with a vertical stretch factor of 3.