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Chapter 1: Problem 2

Explain how each graph is obtained from the graph of \(y=f(x)\) $$ \begin{array}{ll}{\text { (a) } y=f(x)+8} & {\text { (b) } y=f(x+8)} \\\ {\text { (c) } y=8 f(x)} & {\text { (d) } y=f(8 x)} \\ {\text { (e) } y=-f(x)-1} & {\text { (f) } y=8 f\left(\frac{1}{8} x\right)}\end{array} $$

Short Answer

Expert verified
Translation, stretch, or reflection transforms; see steps for specifics.

Step by step solution

01

Vertical Translation

To obtain the graph of \( y = f(x) + 8 \) from \( y = f(x) \), perform a vertical translation of the graph upwards by 8 units. This means every y-coordinate on the original graph is increased by 8.
02

Horizontal Translation

To obtain the graph of \( y = f(x+8) \) from \( y = f(x) \), perform a horizontal translation of the graph to the left by 8 units. This moves each point on the graph 8 units in the negative x-direction.
03

Vertical Stretch

To obtain the graph of \( y = 8f(x) \) from \( y = f(x) \), apply a vertical stretch by a factor of 8. This multiplies the y-coordinate of each point on the original graph by 8, making it taller.
04

Horizontal Compression

To obtain the graph of \( y = f(8x) \) from \( y = f(x) \), compress the graph horizontally by a factor of 8. This means each x-coordinate is divided by 8, making the graph narrower.
05

Reflection and Vertical Translation

To obtain the graph of \( y = -f(x) - 1 \) from \( y = f(x) \), first reflect the graph over the x-axis to get \( -f(x) \), then translate it downward by 1 unit.
06

Vertical Stretch and Horizontal Stretch

To obtain the graph of \( y = 8 f\left(\frac{1}{8} x\right) \) from \( y = f(x) \), first stretch the graph vertically by a factor of 8 and then stretch it horizontally by a factor of 8. This changes each y-coordinate by multiplying by 8 and each x-coordinate by multiplying by 8.

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Key Concepts

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

Vertical Translation
A vertical translation involves shifting a graph up or down along the y-axis. This transformation affects the y-values of the function but leaves the x-values unchanged. When we have a function like \(y = f(x) + 8\), this indicates a vertical translation. Here, the 8 adds to each y-coordinate of the original function \(y = f(x)\).
  • This means that every point on the graph moves upwards by 8 units.
  • No changes occur in the horizontal placement of the graph.
  • This transformation is great for adjusting the height of the graph without altering its shape or direction.
In the context of graph transformations, adding a constant to a function can easily be visualized as a lift or a fall of the whole graph, depending on whether the constant is positive or negative.
Horizontal Translation
Horizontal translation shifts a graph left or right along the x-axis. This transformation modifies the x-values while keeping the y-values unchanged. For example, when dealing with the transformation \(y = f(x + 8)\), a horizontal translation occurs.
  • The addition inside the parentheses moves the graph to the left by 8 units. Even though it seems counterintuitive, adding to the x-value inside the function shifts the graph left, while subtracting would shift it right.
  • The graph's shape remains intact; only the position changes.
Understanding horizontal translation is key in predicting how a function's graph alters its location along the x-axis.
Vertical Stretch
Vertical stretch involves elongating a graph vertically, making it taller or shorter. This transformation multiplies all the y-values by a constant factor \(k\) without changing the x-values. For instance, in \(y = 8f(x)\), a vertical stretch is applied.
  • The factor of 8 stretches every y-coordinate by multiplying it by 8, thus expanding the graph vertically.
  • This makes the peaks and valleys of the graph more pronounced.
  • The deformation maintains the overall shape of the graph, creating a taller version of the original.
Vertical stretching is an essential tool when altering the amplitude of the graph of a function, which might be useful in visually emphasizing certain features.
Horizontal Compression
Horizontal compression happens by squeezing a graph towards the y-axis. This transformation affects the x-values by dividing them by a constant factor, while the y-values remain unchanged. For example, in the transformation \(y = f(8x)\), a horizontal compression is applied.
  • Every x-coordinate on the graph is divided by 8, making the graph appear narrower.
  • This adjustment contracts the graph's width, emphasizing the central portion of the graph.
  • The relative height of the graph does not change; only its horizontal stretch is affected.
Horizontal compression is pivotal when trying to visualize functions more compactly, allowing you to focus on specific x-value intervals more easily.

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Most popular questions from this chapter

Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. \(f(x)=\frac{x}{x^{2}+1}\)

Assume that \(f\) is a one-to-one function. (a) If \(f(6)=17,\) what is \(f^{-1}(17) ?\) (b) If \(f^{-1}(3)=2,\) what is \(f(2) ?\)

If \(f\) and \(g\) are both even functions, is the product \(f g\) even? If \(f\) and \(g\) are both odd functions, is \(f g\) odd? What if \(f\) is even and \(g\) is odd? Justify your answers.

Express the function in the form \(f \circ g\) \(v(t)=\sec \left(t^{2}\right) \tan \left(t^{2}\right)\)

Suppose that the graph of \(y=\log _{2} x\) is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 \(\mathrm{ft}\) ?
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