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Chapter 2: Problem 33

Exer. 33-40: Explain how the graph of the function compares to the graph of \(y=f(x) .\) For example, for the equation \(y=2 f(x+3),\) the graph of \(f\) is shifted 3 units to the Ieft and stretched vertically by a factor of \(2 .\) $$y=f(x-2)+3$$

Short Answer

Expert verified
The graph shifts 2 units right and 3 units up.

Step by step solution

01

Identify Horizontal Shifts

The function given is \( y=f(x-2)+3 \). The \( (x-2) \) inside the function indicates a horizontal shift. Specifically, it means that the graph of \( f(x) \) is shifted 2 units to the right.
02

Identify Vertical Shifts

The addition of \(+3\) outside the function indicates a vertical shift. This means the graph of the function is shifted upward by 3 units.
03

Analyze Collective Transformation

Combine the effects of the transformations noted in the previous steps. The graph of \( y=f(x-2)+3 \) is the graph of \( y=f(x) \) moved 2 units to the right and 3 units upward. There are no vertical or horizontal stretches or reflections involved in this transformation.

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Key Concepts

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

Horizontal Shift
A horizontal shift affects the position of a graph along the x-axis. This transformation occurs when there are adjustments to the variable inside the function. In the expression \( y = f(x - 2) \), the \((x - 2)\) indicates a horizontal shift. To determine the direction of the shift, look at the sign in front of the number.
  • If it is \(x - c\), the graph shifts to the right by \(c\) units.
  • If it is \(x + c\), the graph shifts to the left by \(c\) units.
In our example, \( (x-2) \) moves the graph of \(y = f(x)\) 2 units to the right. With horizontal shifts, the shape of the graph remains unchanged; only its position is altered.
Vertical Shift
A vertical shift involves moving the graph up or down along the y-axis. This occurs when a constant is added or subtracted after the function expression. For \( y = f(x-2) + 3 \), the \(+3\) indicates a vertical shift.
  • If you add a number, \(+ c\), the graph moves up by \( c \) units.
  • If you subtract a number, \(- c\), the graph moves down by \(c\) units.
In our case, the \(+3\) shifts the graph of \(y = f(x)\) upwards by 3 units. Vertical shifts do not affect the shape or orientation of the graph, only its vertical placement is adjusted.
Graph Translation
Graph translation refers to the overall movement of a graph in both horizontal and vertical directions, as seen with transformations like \( y = f(x-2) + 3 \). This combines both horizontal and vertical shifts into one equation, effectively translating the graph without altering its shape or size.For the equation \( y = f(x-2) + 3 \), the graph of \( y = f(x) \) moves 2 units rightward and 3 units upward. This result is from combining:
  • A horizontal shift caused by \( x - 2 \)
  • A vertical shift from \( +3 \)
Graph translation is a fundamental visual transformation for understanding changes in function representation without distortion.
Precalculus Functions
Precalculus functions form the foundation for understanding basic function transformations like shifts and translations. Precalculus functions typically include polynomial, exponential, logarithmic, and trigonometric functions, all of which can undergo transformations to better understand their behavior and properties.In studying function transformations such as \( y = f(x-2) + 3 \), precalculus provides insight into aspects like:
  • Recognizing the basic shape of the graph before transformation.
  • Understanding how transformations affect key characteristics like intercepts and asymptotes.
Mastering these concepts prepares students to advance into more complex areas of calculus and applied mathematics.

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Most popular questions from this chapter

Crest vertical curves When engineers plan highways, they must design hills so as to ensure proper vision for drivers. Hills are referred to as crest vertical curves. Crest vertical curves change the slope of a highway. Engineers use a parabolic shape for a highway hill, with the vertex located at the top of the crest. Two roadways with different slopes are to be connected with a parabolic crest curve. The highway passes through the points \(A(-800,-48), B(-500,0), C(0,40)\) \(D(500,0),\) and \(E(800,-48),\) as shown in the figure. The roadway is linear between \(A\) and \(B\), parabolic between \(B\) and \(\vec{D},\) and then linear between \(D\) and \(E\) (a) Find a piecewise-defined function \(f\) that models the roadway between the points \(A\) and \(E\) (b) Graph \(f\) in the viewing rectangle \([-800,800,100]\) by \([-100,200,100]\)

Sketch the graph of \(f\) $$f(x)=\left\\{\begin{array}{ll} -1 & \text { if } x \text { is an integer } \\ -2 & \text { if } x \text { is not an integer } \end{array}\right.$$

Find a composite function form for \(y\). $$y=\frac{1}{\left(x^{2}+3 x-5\right)^{3}}$$

It has been estimated that 1000 curies of a radioactive substance introduced at a point on the surface of the open sea would spread over an area of \(40,000\) \(\mathrm{km}^{2}\) in 40 days. Assuming that the area covered by the radioactive substance is a linear function of time \(t\) and is always circular in shape, express the radius \(r\) of the contamination as a function of \(t\)

Graph \(f\) in the viewing rectangle \([-12,12]\) by \([-8,8] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same viewing rectangle. $$f(x)=x^{3}-5 x ; \quad g(x)=\left|x^{3}-5 x\right|$$
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