Problem 67 If $(r, 0)$ is an $x$ -inter... [FREE SOLUTION]

91影视

A Graphical Approach to Precalculus with Limits

John Hornsby, Margaret L. Lial, Gary Rockswold

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 2: Problem 67

If $(r, 0)$ is an $x$ -intercept of the graph of $y=f(x),$ what statement can be made about an $x$ -intercept of the graph of each function? (Hint: Make a sketch.) (a) $y=-f(x)$ (b) $y=f(-x)$ (c) $y=-f(-x)$

Short Answer

Expert verified

(a) (r, 0), (b) (-r, 0), (c) (-r, 0).

Step by step solution

Understanding x-intercepts

Recall that an x-intercept of a function is a point where the graph of the function crosses the x-axis. This means that for a point $(x, y)$ to be an x-intercept, the y-coordinate must be zero, i.e., if $(r, 0)$ is the x-intercept of $y=f(x)$, then $f(r) = 0$.

Evaluate y = -f(x)

For the function $y = -f(x)$, we can substitute the x-intercept $(r, 0)$ from $y=f(x)$. Since $f(r) = 0$, we have $-f(r) = -0 = 0$. Thus, $(r, 0)$ remains an x-intercept.

Evaluate y = f(-x)

For the function $y = f(-x)$, we consider what happens to x-intercepts. If $f(r) = 0$, the point $(r, 0)$ is an x-intercept of $y=f(x)$. We need $f(-r) = 0$ for $(r, 0)$ to be an x-intercept of $y=f(-x)$, giving us $(-r, 0)$ as the x-intercept.

Evaluate y = -f(-x)

For the function $y = -f(-x)$, we observe that it combines the transformations from steps 2 and 3. Substitute $f(-r) = 0$ for $(r, 0)$ to be an x-intercept of $y=-f(-x)$. This results in $f(-r) = 0 = 0$, and hence $(-r, 0)$ is the x-intercept.

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Key Concepts

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

Graph Transformations

Graph transformations can significantly change the appearance and position of a function's graph. Transformations help us understand how the graph behaves when we apply various operations like reflection or translation to a function.

Reflection: One of the most common transformations, reflection, occurs when we apply a negative sign to a function, resulting in the graph flipping over a particular axis.
Translation: Involves shifting the graph up, down, left, or right. For example, adding a constant to the function shifts the graph vertically.
Stretching and Compressing: Changing the scale of the graph by multiplying the function by a constant. This adjusts the width or height of the graph.

In our exercise, different transformations are applied:

For $ y = -f(x) $, the graph reflects over the x-axis.
For $ y = f(-x) $, the graph reflects over the y-axis, flipping any x-intercepts to its opposite coordinate.
For $ y = -f(-x) $, both x and y reflections occur, creating a vertical and horizontal flip.

Understanding these transformations helps deduce how x-intercepts are affected by alterations in their graph representations.

Precalculus Functions

Precalculus functions serve as the building blocks for more complex mathematical concepts, preparing students for calculus. These functions form a significant foundation, focusing on properties and their transformations. Key precalculus functions encompass:

Polynomial Functions: Such as quadratic and cubic functions appearing often in transformations.
Exponential and Logarithmic Functions: Explain growth patterns and log-based calculations.
Trigonometric Functions: Define periodic phenomena and include sine, cosine, etc.

By understanding precalculus functions, students are ready to tackle transformations explored in this exercise. Interpreting how transformations affect x-intercepts provides a deeper insight into function behavior.

Function Evaluation

Function evaluation involves determining the output of a function for a given input, usually presented as $ f(x) $. It's a crucial part of analyzing how functions transform and predict changes in graph properties, like x-intercepts.Evaluating a function steps include:

Substitute the input value into the function, such as placing $ x = r $.
Solve the function equation to obtain the output, i.e., $ y = f(x) $.

In the context of x-intercepts and the transformations within our exercise, evaluating functions involves:

For $ y = -f(x) $, ensuring $ f(r) = 0 $ results in the same x-intercept $ (r, 0) $.
For $ y = f(-x) $, check if $ f(-r) = 0 $ for $ (-r, 0) $ to be an x-intercept.
For $ y = -f(-x) $, combine steps to achieve $ (-r, 0) $ as the x-intercept.

By mastering function evaluation, students can confidently navigate various transformations, deepening their understanding of x-intercepts and broader function behavior.

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91影视

If \((r, 0)\) is an \(x\) -intercept of the graph of \(y=f(x),\) what statement can be made about an \(x\) -intercept of the graph of each function? (Hint: Make a sketch.) (a) \(y=-f(x)\) (b) \(y=f(-x)\) (c) \(y=-f(-x)\)

Short Answer

Step by step solution

Understanding x-intercepts

Evaluate y = -f(x)

Evaluate y = f(-x)

Evaluate y = -f(-x)

Key Concepts

Graph Transformations

Precalculus Functions

Function Evaluation

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