Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

2. Graphs of Equations

Graphs and Coordinates

College Algebra

2. Graphs of Equations

Graphs and Coordinates

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1:06 minutes

Problem 90

Textbook Question

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] y = 2(x + 2)^2 + 5(x + 2) - 3

Verified step by step guidance

Rewrite the equation in terms of a single variable by setting y = 0 to find the x-intercepts: 0 = 2(x + 2)^2 + 5(x + 2) - 3.

Let u = x + 2, then substitute u into the equation: 0 = 2u^2 + 5u - 3.

Use the quadratic formula to solve for u: u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a = 2, b = 5, and c = -3.

Calculate the discriminant: b^2 - 4ac = 5^2 - 4(2)(-3).

Substitute the values into the quadratic formula to find the values of u, then solve for x by substituting back x = u - 2.

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Key Concepts

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

X-Intercepts

X-intercepts are the points where a graph crosses the x-axis, which occur when the value of y is zero. To find the x-intercepts of an equation, you set the equation equal to zero and solve for x. This is crucial for understanding the behavior of the graph and how it relates to the equation.

Factoring and Quadratic Equations

Factoring is a method used to simplify quadratic equations, which are polynomials of degree two. The given equation can be expressed in a standard form, allowing for easier identification of roots. Understanding how to factor or apply the quadratic formula is essential for finding x-intercepts.

Graph Matching

Graph matching involves comparing the characteristics of the derived x-intercepts and the shape of the graph to identify the correct representation. Each graph has unique features such as vertex, direction, and intercepts, which help in associating the equation with its corresponding graph accurately.