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Chapter 1: Problem 90

CAPSTONE Match the equation or equations with the given characteristic. (i) \(y = 3x^3 - 3x\) (ii) \(y = (x+3)^2\) (iii) \(y = 3x - 3\) (iv) \(y = \sqrt[3]{x}\) (v) \(y =3x^2 + 3\) (vi) \(y = \sqrt{x+3}\) (a) Symmetric with respect to the \(y\)-axis (b) Three \(x\)-intercepts (c) Symmetric with respect to the \(x\)-axis (d) \((-2, 1)\) is a point on the graph (e) Symmetric with respect to the origin (f ) Graph passes through the origin

Short Answer

Expert verified
(i) matches with characteristics (b) and (e), (iii) matches with (f), (iv) matches with (e), and (vi) matches with (d). Functions (ii) and (v) don't match with any characteristic.

Step by step solution

01

- Analyze each function

First assess each function individually. \n (i) is a cubic function, which is symmetric with respect to the origin and crosses the x-axis three times. Hence, (i) matches with characteristics (b) and (e)\n (ii) is a quadratic function shifted to left by 3 units. It has a symmetry with respect to a vertical line \(x = -3\) (not the y-axis). Thus, (ii) doesn鈥檛 match with any characteristic.\n (iii) is a linear function with a y-intercept as -3. It passes through the origin. So, (iii) matches with (f).\n (iv) is a cubic root function, which is symmetric with respect to the origin. Thus, (iv) matches with (e).\n (v) is a quadratic function, which always results in unwanted symmetry. It doesn鈥檛 match with any characteristic.\n (vi) is a square root function shifted left by 3 units. The point (-2,1) lies on it. Thus, (vi) matches with (d).
02

- Review and verify

Before final conclusion, verify each matching from step 1 again. It can be confirmed that (i) matches with (b) and (e), (ii) doesn't match with any characteristic, (iii) matches with (f), (iv) matches with (e), (v) doesn't match with any characteristic, and (vi) matches with (d).

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Key Concepts

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

Symmetry in Graphs
Understanding symmetry in graphs is essential in precalculus, as it allows you to predict the behavior of a function and its graph. Graphs can be symmetric about the y-axis, x-axis, or the origin. If a graph is symmetric about the y-axis, flipping it over the y-axis will produce the same graph, which generally means for every point \(x, y\), there is a corresponding point \( -x, y\). Functions like \(y = x^2\) show this kind of symmetry.

Graphs symmetric about the x-axis, which is less common in functions due to the vertical line test, would imply that for every point \(x, y\), there is a point \(x, -y\). However, for a function to be symmetric about the origin, rotating it 180 degrees around the origin should leave the graph unchanged. This means for every point \(x, y\), there is also \( -x, -y\). For example, the function \(y = x^3\) displays origin symmetry.
X-intercepts and Y-intercepts
The points where a graph crosses the x-axis and y-axis are known as x-intercepts and y-intercepts, respectively. These intercepts offer valuable information about the function's behavior and are foundational in graph sketching.

The x-intercept(s) are the point(s) where the function's output is zero (\(y=0\)), and solving for \(x\) at these points provides the x-intercept(s). For instance, a quadratic function such as \(y = x^2 - 4\) has two x-intercepts at \(x = 2\) and \(x = -2\). On the other hand, the y-intercept is where the graph crosses the y-axis, so the x-value is zero (\(x=0\)), and solving for \(y\) will give you the y-intercept. For the same quadratic function, the y-intercept is \(y = -4\).

Knowing how to find these intercepts is crucial since they help to construct the graph of the function and understand its zeros and initial value, respectively.
Transformation of Functions
Transformation of functions is a powerful concept in precalculus that involves shifting, reflecting, stretching, or compressing the graph of a function. These transformations allow us to modify a base function into a new one that better fits a given situation or equation.

Generally speaking, a shift to the right or left involves adding or subtracting a constant from the \(x\)-variable, respectively. A shift up or down involves adding or subtracting a constant from the function's output (the \(y\) value). For example, the transformation of \(y = x^2\) to \(y = (x - 2)^2 + 3\) involves shifting the parabola 2 units to the right and 3 units up.

Reflections are another type of transformation where the graph is flipped over an axis or the origin. A negative in front of the function reflects it across the x-axis, while a negative inside the function reflects it across the y-axis.

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Most popular questions from this chapter

In Exercises 87-92, use the functions given by \(f(x) = \frac{1}{8}x - 3\) and \(g(x) = x^3\) to find the indicated value or function. \((f \circ g)^{-1}\)

COLLEGE ENROLLMENT The Pennsylvania State University had enrollments of 40,571 students in 2000 and 44,112 students in 2008 at its main campus in University Park, Pennsylvania. (Source: Penn State Fact Book) (a) Assuming the enrollment growth is linear, find a linear model that gives the enrollment in terms of the year \(t\) where \(t=0\) corresponds to 2000. (b) Use your model from part (a) to predict the enrollments in 2010 and 2015. c) What is the slope of your model? Explain its meaning in the context of the situation.

MAKE A CONJECTURE Plot the points \( (2, 1) \), \( (-3, 5) \) and \( (7, -3) \) on a rectangular coordinate system. Then change the sign of the \( x \)-coordinate of each point and plot the three new points on the same rectangular coordinate system. Make a conjecture about the location of a point when each of the following occurs. (a) The sign of the \( x \)-coordinate is changed. (b) The sign of the \( y \)-coordinate is changed. (c) The signs of both the \( x \)- and \( y \)-coordinates are changed.

In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. \(f(x) = x^3 - 3x^2 + 2\)

RESISTANCE In Exercises 77 and 78, use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross-sectional area. If #28 copper wire (which has a diameter of 0.0126 inch) has a resistance of 66.17 ohms per thousand feet, what length of #28 copper wire will produce a resistance of 33.5 ohms?
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