Question

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

Explanation:

Step1: Identify the parent function

The parent function is \( y = |x| \), which is a V - shaped graph with vertex at \((0,0)\), opening upwards, with slopes \(1\) (for \(x\geq0\)) and \(- 1\) (for \(x < 0\)).

Step2: Analyze the transformations

For the function \( y=5|x| + 2\):

• The coefficient \(5\) (a vertical stretch by a factor of \(5\)): This makes the graph steeper. For the parent function \(y = |x|\), when \(x = 1\), \(y = 1\); for \(y=5|x|\), when \(x = 1\), \(y=5\times1 = 5\); when \(x=- 1\), \(y = 5\times|-1|=5\).
• The \(+2\) (a vertical shift up by \(2\) units): This moves the vertex of the graph from \((0,0)\) to \((0,2)\).

Step3: Find key points

• When \(x = 0\): \(y=5|0|+2=2\), so the vertex is at \((0,2)\).
• When \(x = 1\): \(y=5|1|+2=5 + 2=7\), so the point is \((1,7)\).
• When \(x=-1\): \(y=5|-1|+2=5 + 2=7\), so the point is \((-1,7)\).
• When \(x = 2\): \(y=5|2|+2=10 + 2=12\) (but since the grid goes up to \(y = 10\), we can also use the slope. The slope for \(x\geq0\) is \(5\) (rise over run, for each unit increase in \(x\), \(y\) increases by \(5\)) and for \(x<0\) the slope is \(- 5\) (for each unit decrease in \(x\) (i.e., moving left), \(y\) increases by \(5\)).

Step4: Plot the points and draw the graph

• Plot the vertex \((0,2)\).
• Plot the points \((1,7)\) and \((-1,7)\).
• Since the graph is symmetric about the \(y\) - axis (because it's a function of \(|x|\)), we can draw two lines:
• For \(x\geq0\), the line passes through \((0,2)\) and \((1,7)\). The slope \(m=\frac{7 - 2}{1-0}=5\), so the equation of the line for \(x\geq0\) is \(y - 2=5(x - 0)\) or \(y = 5x+2\).
• For \(x<0\), the line passes through \((0,2)\) and \((-1,7)\). The slope \(m=\frac{7 - 2}{-1 - 0}=-5\), so the equation of the line for \(x<0\) is \(y - 2=-5(x - 0)\) or \(y=-5x + 2\).
• Draw the two lines to form the V - shaped graph with vertex at \((0,2)\), passing through \((1,7)\) and \((-1,7)\) (and other points following the slope), opening upwards, and steeper than the parent function \(y = |x|\).

(To actually draw on the given grid: Start at \((0,2)\). For the right side (\(x\geq0\)), from \((0,2)\), move 1 unit right and 5 units up to \((1,7)\) (but since the grid has \(y\) - axis up to \(10\), we can also use the fact that the slope is \(5\), so the rate of change is steep). For the left side (\(x<0\)), from \((0,2)\), move 1 unit left and 5 units up to \((-1,7)\). Then draw the two line segments from the vertex through these points, extending them as needed, keeping the V - shape.)

Answer:

The graph is a V - shaped graph (absolute - value graph) with vertex at \((0,2)\), passing through \((-1,7)\) and \((1,7)\), with a vertical stretch by a factor of \(5\) and a vertical shift up by \(2\) units from the parent function \(y = |x|\). (To draw it, plot the vertex \((0,2)\), then points like \((-1,7)\) and \((1,7)\), and draw the two line segments from the vertex through these points, following the slope of \(5\) (for \(x\geq0\)) and \(-5\) (for \(x < 0\)).)