Textbook Question
Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=(1/2)(x-2)2+4
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 19
Graph the following on the same coordinate system.
(a) y = x2
(b) y = 3x2
(c) y = 1/3x2
(d) How does the coefficient of x2 affect the shape of the graph?
Verified step by step guidance1
Identify the base function for all graphs, which is the quadratic function \(y = x^2\). This is a parabola opening upwards with its vertex at the origin \((0,0)\).
For the function \(y = 3x^2\), recognize that the coefficient 3 is greater than 1, which causes the parabola to become narrower (steeper) compared to \(y = x^2\). Plot points by substituting values of \(x\) and calculating \(y\) to see this effect.
For the function \(y = \frac{1}{3}x^2\), note that the coefficient \(\frac{1}{3}\) is between 0 and 1, which makes the parabola wider (less steep) than \(y = x^2\). Again, plot points by substituting values of \(x\) to observe this change.
Graph all three functions on the same coordinate system by plotting several points for each and drawing smooth parabolas through these points, ensuring the vertex is at the origin for all.
Analyze how the coefficient of \(x^2\) affects the shape: coefficients greater than 1 make the parabola narrower, coefficients between 0 and 1 make it wider, and a negative coefficient would reflect it across the x-axis (not in this problem).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Functions and Their Graphs
A quadratic function is a polynomial of degree two, typically written as y = ax^2 + bx + c. Its graph is a parabola, which is symmetric and opens upward if a > 0. Understanding the basic shape of y = x^2 helps in comparing how changes in coefficients affect the graph.
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Graphs of Logarithmic Functions
Effect of the Leading Coefficient on the Parabola
The coefficient 'a' in y = ax^2 controls the parabola's width and direction. If |a| > 1, the parabola becomes narrower (steeper), and if 0 < |a| < 1, it becomes wider (flatter). A positive 'a' opens upward, while a negative 'a' opens downward.
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5:28Horizontal Parabolas
Graphing Multiple Functions on the Same Coordinate System
Plotting multiple functions together allows comparison of their shapes and behaviors. By graphing y = x^2, y = 3x^2, and y = (1/3)x^2 on the same axes, one can visually analyze how different coefficients affect the parabola's width and steepness.
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Guided course
05:10Graphs & the Rectangular Coordinate System
Related Practice
Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation. x2 - 2 > x
Textbook Question
Use synthetic division to perform each division. (x4 - 3x3 - 4x2 + 12x) / x-2
Textbook Question
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first.
Textbook Question
Match each statement with its corresponding graph in choices A–D. In each case, k > 0. y varies directly as the second power of x. (y=kx2)
Textbook Question
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.