Question
\( \begin{array}{ll}\text { Graph rach function by translating } y=|x| \\ \text { 32.) } y=|x|-5 & \text { 33.) } y=|x+7|\end{array} \)
Ask by Boone Ball. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To graph \( y = |x| - 5 \), start with the V-shaped graph of \( y = |x| \) and move it down 5 units, placing the vertex at \( (0, -5) \). For \( y = |x+7| \), take the same V-shaped graph and shift it 7 units to the left, placing the vertex at \( (-7, 0) \).
Solution
1. For \( y = |x| - 5 \):
- Start with the graph of \( y = |x| \), which has its vertex at \( (0,0) \).
- The expression \(-5\) translates the graph downward by 5 units.
- Thus, the vertex moves to \( (0,-5) \), and the graph opens upward producing a V-shape with the vertex at \( (0,-5) \).
2. For \( y = |x+7| \):
- Start with the graph of \( y = |x| \), which has its vertex at \( (0,0) \).
- The expression \(x+7\) inside the absolute value represents a horizontal translation. Since \(x\) is replaced by \(x+7\), the graph shifts 7 units to the left.
- Therefore, the vertex becomes \( (-7,0) \), and the graph still has the V-shape opening upward.
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Simplify this solution Beyond the Answer
To graph \( y = |x| - 5 \), start with the basic V-shape of \( y = |x| \). Then, shift this V-shape downward by 5 units. The vertex will now be at the point (0, -5), and the arms will still rise at the same angles as the original graph. For \( y = |x + 7| \), you start again with the graph of \( y = |x| \). This time, you’ll shift it to the left by 7 units. The new vertex will be at (-7, 0), and the V-shape will open upward with the same slopes, just now centered around -7 on the x-axis. Happy graphing! 🎉
