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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 76
Chapter 3, Problem 76

Consider the following nonlinear system. Work Exercises 75 –80 in order.
y=∣x−1∣y = | x - 1 |
y=x2−4y = x^2 - 4
How is the graph of y = x^2 - 4 obtained by transforming the graph of y=x2y = x^2?

Verified step by step guidance
1
Recall the parent function for the quadratic is given by \(y = x^2\), which is a parabola with vertex at the origin \((0,0)\).
The given function is \(y = x^2 - 4\). Notice that this is the parent function \(y = x^2\) with a constant subtracted.
Subtracting 4 from \(x^2\) means every \(y\)-value of the original parabola is decreased by 4 units.
This results in a vertical shift of the graph downward by 4 units.
Therefore, the graph of \(y = x^2 - 4\) is obtained by shifting the graph of \(y = x^2\) down 4 units, moving the vertex from \((0,0)\) to \((0,-4)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parent Function and Transformations

The parent function y = x^2 is the basic quadratic function whose graph is a parabola centered at the origin. Transformations involve shifting, stretching, or reflecting this graph to produce new functions. Understanding how changes to the equation affect the graph is essential for analyzing y = x^2 - 4.
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Domain & Range of Transformed Functions

Vertical Shifts

A vertical shift moves the graph up or down without changing its shape. In the function y = x^2 - 4, subtracting 4 shifts the entire parabola downward by 4 units. This means every point on y = x^2 moves 4 units lower on the y-axis.
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Shifts of Functions

Graphing Nonlinear Systems

Graphing nonlinear systems involves plotting multiple nonlinear equations to find points of intersection or analyze their behavior. Understanding each graph individually, such as y = |x - 1| and y = x^2 - 4, helps in comparing and solving the system.
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Nonlinear Inequalities
Related Practice
Textbook Question

If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear. (-1, -3), (-5, 12), (1, -11)

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. h(x)=-(x+1)3

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Given functions f and g, find (b)(g∘ƒ)(x)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

ƒ(x)=√x,g(x)=x+3ƒ(x)=√x, g(x)=x+3

Textbook Question

Graph each function. f(x)={∣x∣if x<36−xif x≥3f(x) =\(\begin{cases}\)|x| & \(\text{if }\) x < 3 \\6 - x & \(\text{if }\) x \(\geq\) 3\(\end{cases}\)

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Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.

ƒ(x)=√x,g(x)=x+3ƒ(x)=√x, g(x)=x+3

Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√x, g(x)=x-1