Question

graph the equation.
$y = 5|x| + 3$

Explanation:

Step1: Identify the parent function

The equation \( y = 5|x| + 3 \) is a transformation of the parent absolute - value function \( y=|x| \). The parent function \( y = |x| \) has a V - shape with its vertex at the origin \((0,0)\), and for \( x\geq0 \), \( y = x \); for \( x<0 \), \( y=-x \).

Step2: Analyze the transformations

• Vertical stretch: The coefficient \( 5 \) in front of \( |x| \) means that the graph of \( y = |x| \) is vertically stretched by a factor of \( 5 \). For the parent function, when \( x = 1 \), \( y = 1 \); for \( y=5|x| \), when \( x = 1 \), \( y=5\times|1|=5 \); when \( x=- 1 \), \( y = 5\times|-1| = 5 \).
• Vertical shift: The \( + 3 \) at the end of the equation means that the graph is shifted up by \( 3 \) units. So the vertex of the graph of \( y=5|x|+3 \) will be at \((0,3)\) (since the vertex of \( y = 5|x| \) is at \((0,0)\) and we shift it up by 3 units).

Step3: Find key points

• When \( x = 0 \): Substitute \( x = 0 \) into the equation \( y=5|0|+3=3 \). So the point \((0,3)\) is on the graph (the vertex).
• When \( x = 1 \): \( y=5|1|+3=5 + 3=8 \). So the point \((1,8)\) is on the graph.
• When \( x=-1 \): \( y=5|-1|+3=5 + 3=8 \). So the point \((-1,8)\) is on the graph.
• When \( x = 2 \): \( y=5|2|+3=10 + 3=13 \) (but since our graph only goes up to \( y = 10 \) in the given grid, we can also use the fact that the slope of the right - hand side (\(x\geq0\)) of the absolute - value graph \( y = 5|x|+3 \) is \( 5 \), and the left - hand side (\(x<0\)) has a slope of \(- 5\)).

Step4: Sketch the graph

• Plot the vertex \((0,3)\).
• Plot the points \((1,8)\) and \((-1,8)\).
• Since the absolute - value function has a V - shape, draw two lines: one with a slope of \( 5 \) passing through \((0,3)\) and \((1,8)\) (for \( x\geq0 \)) and one with a slope of \(-5\) passing through \((0,3)\) and \((-1,8)\) (for \( x<0\)). The graph will be a V - shaped graph, vertically stretched, shifted up, with the vertex at \((0,3)\), and passing through points like \((1,8)\) and \((-1,8)\), \((2,13)\) (if we extend), etc.

(Note: Since the problem is to graph the equation, the final answer is the graph with vertex at \((0,3)\), passing through \((1,8)\), \((-1,8)\) and having a V - shape with a vertical stretch factor of 5 and a vertical shift of 3 units up from the parent absolute - value function.)

Answer:

The graph of \( y = 5|x|+3 \) is a V - shaped graph with vertex at \((0,3)\), passing through \((1,8)\) and \((-1,8)\), vertically stretched by a factor of 5 and shifted up 3 units from the parent function \( y = |x|\). To draw it, plot the vertex \((0,3)\), then plot points like \((1,8)\) and \((-1,8)\) and draw two lines with slopes \( 5 \) (for \( x\geq0 \)) and \(-5\) (for \( x < 0\)) through these points.