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Chapter 0: Problem 39

Shifting a Graph Let \(f(x)=x^{2} .\) Graph the functions \(f(x)+1\) \(f(x)-1, f(x)+2,\) and \(f(x)-2 .\) Make a guess about the relationship between the graph of a general function \(f(x)\) and the graph of \(f(x)+c\) for some constant \(c .\) Test your guess on the functions \(f(x)=x^{3}\) and \(f(x)=\sqrt{x}\).

Short Answer

Expert verified
The graph of \( f(x) + c \) is the graph of \( f(x) \) shifted vertically by \( c \) units.

Step by step solution

01

- Graph the base function

First, graph the function \( f(x) = x^2 \) on a coordinate plane. This is a standard parabola that opens upwards.
02

- Graph the shifted functions

Next, graph the following functions based on the base function \( f(x) = x^2 \): \( f(x) + 1 = x^2 + 1 \), \( f(x) - 1 = x^2 - 1 \), \( f(x) + 2 = x^2 + 2 \), \( f(x) - 2 = x^2 - 2 \). Each of these graphs will be a parabola shifted up or down according to the constant term.
03

- Analyze the shifts

By comparing the graphs, note that \( f(x) + c \) shifts the graph of \( f(x) \) vertically by \( c \) units. If \( c \) is positive, the shift is upward; if \( c \) is negative, the shift is downward.
04

- Test the guess with \( f(x) = x^3 \)

Graph \( f(x) = x^3 \) and then graph \( f(x) + 1 = x^3 + 1 \) and \( f(x) - 1 = x^3 - 1 \). Observe that the same vertical shift pattern applies.
05

- Test the guess with \( f(x) = \sqrt{x} \)

Graph \( f(x) = \sqrt{x} \) and then graph \( f(x) + 1 = \sqrt{x} + 1 \) and \( f(x) - 1 = \sqrt{x} - 1 \). Again, observe the vertical shifting.
06

- State the relationship

From the observations, it can be concluded that for any function \( f(x) \), the graph of \( f(x) + c \) is the graph of \( f(x) \) shifted vertically by \( c \) units.

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Key Concepts

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

vertical shift
Understanding the concept of vertical shifts is essential in graphing functions. When we talk about a vertical shift, we are referring to moving the entire graph of a function up or down on the coordinate plane. This is determined by the constant term added to or subtracted from the function. For instance, if you have a function \(f(x)\) and you want to shift it vertically, you simply add or subtract a constant \(c\). If \(c\) is positive, the graph will move up. If \(c\) is negative, the graph will move down. For example, graphing \(f(x) = x^2\) and then graphing \(f(x) + 2 = x^2 + 2\) will show you that the parabola shifts 2 units upward.
function transformation
Function transformation involves various techniques to alter the graph of a function. Vertical shifts are just one type of transformation. Other transformations include horizontal shifts, scaling, reflections, and rotations. Specifically, vertical shifts are a straightforward transformation, where you modify the output value of the function by adding or subtracting a constant. For example, transforming \(f(x) = x^3\) to \(f(x) + 1 = x^3 + 1\) shifts the entire cubic graph up by 1 unit. This technique helps us understand how changes to the function's formula impact its graph.
graphing functions
Graphing functions is a fundamental skill in mathematics. It allows you to visualize the behavior of equations and their transformations. When you graph a function, you plot points that satisfy the equation on a coordinate plane. A standard approach involves first graphing the base function. For example, Start by plotting \(f(x) = x^2\) to see a parabola opening upwards. Then apply transformations like vertical shifts to see how the graph changes. Graphing functions helps in identifying patterns and understanding how modifying equations affects their graphical representation.
coordinate plane
The coordinate plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Every point on this plane is identified by a pair of coordinates \((x, y)\). The coordinate plane is essential for graphing functions. For example, when graphing \(f(x) = \sqrt{x}\), you plot points for various values of x, such as \((0, 0)\), \((1, 1)\), and \((4, 2)\). By connecting these points, you can see the shape of the graph. Using the coordinate plane allows for precise visualization of function transformations like vertical shifts.
constant term
The constant term in a function equation plays a significant role in determining vertical shifts. When we add a constant term \(c\) to a function \(f(x)\), it shifts the graph of the function vertically by \(c\) units. For example, in the function \(f(x) = x^2 + c\), the constant term \(c\) moves the parabola upward if \(c\) is positive, and downward if \(c\) is negative. This relationship holds true for any function, such as polynomials like \(f(x) = x^3\) or radical functions like \(f(x) = \sqrt{x}\). The constant term is key in predicting how a function's graph will shift.

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Most popular questions from this chapter

During the first \(\frac{1}{2}\) hour, the employees of a machine shop prepare the work area for the day's work. After that, they turn out 10 precision machine parts per hour, so the output after \(t\) hours is \(f(t)\) machine parts, where \(f(t)=10\left(t-\frac{1}{2}\right)=10 t-5\) \(\frac{1}{2} \leq t \leq 8 .\) The total cost of producing \(x\) machine parts is \(C(x)\) dollars, where \(C(x)=.1 x^{2}+25 x+200.\) (a) Express the total cost as a (composite) function of \(t.\) (b) What is the cost of the first 4 hours of operation?

Let \(f(x)=\frac{x}{x-2}, g(x)=\frac{5-x}{5+x},\) and \(h(x)=\frac{x+1}{3 x-1} .\) Express the following as rational functions. $$h\left(\frac{1}{x^{2}}\right)$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\left(16 x^{8}\right)^{-3 / 4}$$

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$\sqrt{f(x) g(x)}$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{-x^{3} y}{-x y}$$
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