Question

1. an equation is shown. ( y = 3|x + 2| - 6 ) use the coordinate grid to graph the equation. ©math nation

Explanation:

Step1: Identify the vertex form of absolute value function

The general form of an absolute value function is \( y = a|x - h| + k \), where \((h, k)\) is the vertex. For the given equation \( y = 3|x + 2| - 6 \), we can rewrite it as \( y = 3|x - (-2)| + (-6) \). So, the vertex \((h, k)\) is \((-2, -6)\).

Step2: Determine the slope of the two branches

For \( x \geq -2 \), the equation becomes \( y = 3(x + 2) - 6 = 3x + 6 - 6 = 3x \). The slope here is \( 3 \). For \( x < -2 \), the equation becomes \( y = 3(-(x + 2)) - 6 = -3x - 6 - 6 = -3x - 12 \). The slope here is \( -3 \).

Step3: Plot the vertex and use the slopes to find other points

• Plot the vertex \((-2, -6)\) on the coordinate grid.
• For the right branch ( \( x \geq -2 \) with slope \( 3 \)): From the vertex, move 1 unit to the right (increase \( x \) by 1) and 3 units up (increase \( y \) by 3) to get the point \((-1, -3)\). Move another 1 unit right and 3 units up to get \((0, 0)\), and so on.
• For the left branch ( \( x < -2 \) with slope \( -3 \)): From the vertex, move 1 unit to the left (decrease \( x \) by 1) and 3 units up (increase \( y \) by 3) to get the point \((-3, -3)\). Move another 1 unit left and 3 units up to get \((-4, 0)\), and so on.

Step4: Draw the graph

Connect the points on each branch to form a V - shaped graph with the vertex at \((-2, -6)\), opening upwards (since \( a = 3>0 \)) with the right branch having a slope of \( 3 \) and the left branch having a slope of \( -3 \).

Answer:

The graph of \( y = 3|x + 2| - 6 \) is a V - shaped graph with vertex at \((-2, -6)\), right branch passing through points like \((-1, -3)\), \((0, 0)\) etc. (with slope \( 3 \)) and left branch passing through points like \((-3, -3)\), \((-4, 0)\) etc. (with slope \( -3 \)). (The actual graph is drawn by plotting the vertex and the other points as calculated and connecting them.)