Textbook Question
Find all values of x satisfying the given conditions. y = 2x2 - 3x and y = 2
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 111a
In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = - (x + 1)2 + 4
Verified step by step guidance1
To find the x-intercepts of the graph, set y = 0 in the given equation. The equation becomes 0 = - (x + 1)^2 + 4.
Rearrange the equation to isolate the squared term: (x + 1)^2 = 4.
Take the square root of both sides, remembering to include both the positive and negative roots: x + 1 = ±√4.
Simplify the square root: x + 1 = ±2. This gives two equations: x + 1 = 2 and x + 1 = -2.
Solve each equation for x: For x + 1 = 2, subtract 1 to get x = 1. For x + 1 = -2, subtract 1 to get x = -3. The x-intercepts are x = 1 and x = -3.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
X-Intercept
The x-intercept of a graph is the point where the graph intersects the x-axis. This occurs when the value of y is zero. To find the x-intercept, you set the equation equal to zero and solve for x. In the given equation, this means solving - (x + 1)^2 + 4 = 0.
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Graphing Intercepts
Quadratic Functions
The equation provided is a quadratic function, which is typically in the form y = ax^2 + bx + c. Quadratic functions produce parabolic graphs, which can open upwards or downwards depending on the sign of 'a'. In this case, the negative sign indicates the parabola opens downwards, affecting the location of the x-intercepts.
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Graphing Techniques
Understanding how to graph a quadratic function involves identifying key features such as the vertex, axis of symmetry, and intercepts. The vertex can be found using the formula x = -b/(2a), and the intercepts help in sketching the graph accurately. This knowledge is essential for matching the equation to its corresponding graph.
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Related Practice
Textbook Question
In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = x2 - 2x + 2
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Textbook Question
Solve and graph the solution set on a number line: (2x−3)/4 ≥ 3x/4 + 1/2
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Textbook Question
In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = x2 - 4x - 5
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Textbook Question
Find all values of x satisfying the given conditions. y = 5x2 + 3x and y = 2
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Textbook Question
In Exercises 107–110, use graphs to find each set. [1,3) ∩ (0,4)
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