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Chapter 2: Problem 25

Suppose the point \((8,12)\) is on the graph of \(y=f(x) .\) Find a point on the graph of each function. (a) \(y=f(x+4)\) (b) \(y=f(x)+4\)

Short Answer

Expert verified
(a) (4,12) (b) (8,16)

Step by step solution

01

Identify the transformation for part (a)

For the function transformation given by \(y=f(x+4)\), the term \((x+4)\) indicates a horizontal shift. Specifically, it shifts the graph of \(f(x)\) to the left by 4 units.
02

Apply the transformation for part (a)

To find the new coordinates after the horizontal shift, subtract 4 from the original x-coordinate of the point (8,12). The new x-coordinate is \(8-4=4\). The y-coordinate remains the same, so the new point is \( (4,12) \).
03

Identify the transformation for part (b)

For the function transformation given by \(y=f(x)+4\), the term \(+4\) indicates a vertical shift. Specifically, it shifts the graph of \(f(x)\) upwards by 4 units.
04

Apply the transformation for part (b)

To find the new coordinates after the vertical shift, add 4 to the original y-coordinate of the point (8,12). The new y-coordinate is \(12+4=16\). The x-coordinate remains the same, so the new point is \( (8,16) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Shift
Horizontal shifts involve moving the entire graph of a function left or right along the x-axis. This translation does not alter the shape of the graph, but rather its position relative to the y-axis.

When we have an equation of the form \(y=f(x+c)\), it indicates a horizontal shift:
  • If c is positive, there is a leftward shift by c units.
  • If c is negative, the shift is rightward by the absolute value of c units.

In our exercise, we started with the point (8,12) on the graph of \(y=f(x)\). For the transformation \(y=f(x+4)\), the graph undergoes a leftward shift by 4 units. Therefore, the x-coordinate of the point moves from 8 to 4, with the y-coordinate remaining unchanged. The new point on the graph for \(y=f(x+4)\) is \( (4,12) \).
Vertical Shift
Vertical shifts involve moving the entire graph of a function up or down along the y-axis. Similar to horizontal shifts, this transformation does not change the shape of the graph, merely its position relative to the x-axis.

When we look at an equation like \(y=f(x)+c\), it signals a vertical shift:
  • If c is positive, the graph shifts upwards by c units.
  • If c is negative, the graph shifts downwards by the absolute value of c units.

Referring to our exercise again, we have the point (8,12) on the graph of \(y=f(x)\). For the transformation \(y=f(x)+4\), the graph is shifted upwards by 4 units. Hence, the y-coordinate of the point increases from 12 to 16, while the x-coordinate remains the same. The new point on the graph for \(y=f(x)+4\) is \( (8,16) \).
Graphing Functions
Graphing functions involves plotting points on a coordinate plane that satisfy the function's equation to reveal the overall shape and position of the graph.

Understanding various transformations is key to predicting how these modifications will affect the graph:
  • Horizontal Shifts: Move the graph left or right.
  • Vertical Shifts: Move the graph up or down.
  • Reflections: Flip the graph over a particular axis.
  • Stretching/Shrinking: Alter the graph's width or height.

When you graph a function and apply transformations step-by-step, it's easier to understand these changes. For example, starting with a known point makes it straightforward to see the resulting position after shifts.
By using the exercise's initial point (8,12) and applying a horizontal shift to get (4,12) and a vertical shift to get (8,16), you can visualize how the graph moves in the coordinate plane with these transformations.

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