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Chapter 2: Problem 13

Describing Transformations Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\) (a) \(y=-f(x)+5\) (b) \(y=3 f(x)-5\)

Short Answer

Expert verified
(a) Reflect over x-axis and shift 5 up. (b) Stretch vertically by 3, shift 5 down.

Step by step solution

01

Understanding Vertical Reflections

The term \(-f(x)\) indicates a reflection across the x-axis. This means that every point on the graph of \(f(x)\) will be reflected to the opposite side of the x-axis. If a point on \(f\) was above the x-axis, it will now be below by the same distance, and vice versa.
02

Apply Vertical Shift for (a)

The term \(+5\) in \(y=-f(x)+5\) corresponds to a vertical shift. Specifically, this transformation shifts the graph 5 units upward. So, after reflecting the graph across the x-axis, shift the entire graph upward by 5 units.
03

Reflection and Shift for Part (a)

For the function \(y=-f(x)+5\), apply the reflection over the x-axis and then shift the resulting graph 5 units up. This completes the transformation for part (a).
04

Understanding Vertical Stretch

The coefficient \(3\) in \(3f(x)\) represents a vertical stretch of the graph by a factor of 3. This means that the distance from any point on the graph to the x-axis will be multiplied by 3, making the graph taller if \(f(x)\) is positive, or more negative if \(f(x)\) is negative.
05

Apply Vertical Shift for (b)

The term \(-5\) in \(y=3f(x)-5\) indicates a vertical shift downward by 5 units. After stretching the graph vertically, shift every point downward by 5 units.
06

Stretch and Shift for Part (b)

For the function \(y=3f(x)-5\), apply a vertical stretch by a factor of 3 and then shift the entire graph 5 units down. This completes the transformation for part (b).

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Key Concepts

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

Vertical Reflection
A vertical reflection in mathematics involves flipping the graph of a function over the x-axis. Imagine placing a mirror along the x-axis; the graph will mirror itself on the opposite side. If you have the function \(f(x)\), then \(-f(x)\) represents its vertical reflection. Here鈥檚 how it works:
  • Every point (x, y) on the original graph is flipped to (x, -y).
  • If a point was above the x-axis, it will now be below at the same distance.
  • Similarly, if a point was below, it will move above, mirroring across the x-axis.
To apply this transformation, identify points of interest on \(f(x)\) and simply negate the y-values. This transformation is crucial when dealing with functions that require flipping over the x-axis for analysis or graphical presentation.
Vertical Shift
A vertical shift changes the function's position along the y-axis. When analyzing the function \(y = -f(x)+5\), the term \(+5\) indicates a vertical shift upwards by 5 units. Here鈥檚 the process for understanding vertical shifts:
  • If the term is positive (e.g., +5), shift the graph upward by that many units.
  • If the term is negative (e.g., -5), shift the graph downward.
  • Vertical shifts do not affect the shape of the graph, only its position on the y-axis.
In step-by-step transformations, you would first perform other transformations (like a vertical reflection) and then apply the vertical shift to the whole graph. For example, in \(y = -f(x) + 5\), reflect the graph and then shift it five units up.
Vertical Stretch
A vertical stretch involves modifying the function's amplitude without altering its overall shape. When you multiply \(f(x)\) by a factor larger than 1, you stretch it away from the x-axis. For instance, in \(y = 3f(x)\), each point on the graph is extended three times further from the x-axis:
  • The distance from each point to the x-axis increases by the factor, here 3 times.
  • If \(f(x)\) is positive, the graph becomes taller. If it's negative, it stretches downwards.
  • This transformation doesn鈥檛 shift the graph left or right, only alters the vertical dimension.
After applying the vertical stretch, any additional transformations can be applied, such as vertical shifts, to complete the transformation process. For \(y = 3f(x) - 5\), stretch first, then shift down by 5 units.

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Most popular questions from this chapter

DISCUSS DISCOVER: Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $$g(x)=\sqrt{f(x)}$$ where \(f(x) \geq 0\) for all \(x .\) Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, x^{2}\right)\) on the graph of the parabola \(y=x^{2}\) Express \(g\) as a function of \(x\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.

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