Domanda del libro di testo
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (4−2x)/(3x+4)≤0
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 4, Problema 49
Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x2 but with the given point as the vertex. (5, 3)
Guida verificata passo dopo passo1
Recall that the vertex form of a parabola is given by the equation \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola and \(a\) determines the shape (width and direction) of the parabola.
Identify the value of \(a\) from the given function \(f(x) = 2x^2\). Here, \(a = 2\), which means the parabola opens upward and is narrower than the standard parabola \(y = x^2\).
Use the given vertex point \((5, 3)\) to substitute \(h = 5\) and \(k = 3\) into the vertex form equation, so it becomes \(y = 2(x - 5)^2 + 3\).
This new equation represents a parabola with the same shape as \(f(x) = 2x^2\) but shifted so that its vertex is at \((5, 3)\).
You can verify the correctness by expanding the vertex form and comparing it to the standard form or by plotting the graph to see the vertex at the correct point.
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Vertex Form of a Quadratic Function
The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex and understand the graph's shape and position.
Video consigliato:
08:07
Vertex Form
Effect of the 'a' Coefficient on Parabola Shape
The coefficient 'a' in a quadratic function affects the parabola's width and direction. If |a| > 1, the parabola is narrower; if 0 < |a| < 1, it is wider. A positive 'a' opens upward, while a negative 'a' opens downward.
Video consigliato:
5:28Horizontal Parabolas
Using a Given Vertex to Write the Equation
When given a vertex and the shape of a parabola, substitute the vertex coordinates (h, k) into the vertex form and use the known 'a' value to write the equation. This ensures the parabola has the correct shape and position.
Video consigliato:
04:27Finding Equations of Lines Given Two Points
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