Ch. R - Algebra Review

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. R - Algebra Review

Problema 61

Capitolo 1, Problema 61

Graph each function. See Examples 6–8. g(x) = (x - 4)²

Guida verificata passo dopo passo

Recognize that the function given is a quadratic function in the form \(g(x) = (x - 4)^2\), which is a transformation of the basic parabola \(y = x^2\).

Identify the horizontal shift by noting the expression \((x - 4)\) inside the square. This means the graph of \(y = x^2\) is shifted 4 units to the right.

Determine the vertex of the parabola. Since the basic parabola \(y = x^2\) has its vertex at \((0,0)\), the vertex of \(g(x) = (x - 4)^2\) will be at \((4,0)\).

Understand the shape and direction of the parabola. Because the coefficient of the squared term is positive (implicitly 1), the parabola opens upwards.

To graph the function, plot the vertex at \((4,0)\), then choose values of \(x\) around 4 (such as 3 and 5) to calculate corresponding \(g(x)\) values, and plot these points to sketch the parabola.

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Graphing Quadratic Functions

A quadratic function is a polynomial of degree two, typically written as f(x) = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and direction of the parabola is essential for graphing functions like g(x) = (x - 4)².

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Percorso guidato

6:36

Quadratic Formula

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. For g(x) = (x - 4)², the vertex is at (4, 0). This form makes it easy to identify the vertex and graph the parabola by shifting the basic parabola y = x² horizontally and vertically.

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Percorso guidato

6:36

Quadratic Formula

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the graph of a function. In g(x) = (x - 4)², the graph of y = x² is shifted 4 units to the right. Recognizing these transformations helps in quickly sketching the graph without plotting many points.

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Percorso guidato

4:22

Domain and Range of Function Transformations

Pratica correlata

Domanda del libro di testo

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6. center (√2, √2), radius √2

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Domanda del libro di testo

For each function, find (a) ƒ(2) and (b) ƒ(-1). See Example 7. ƒ = {(-1, 3), (4, 7), (0, 6), (2, 2)}

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Domanda del libro di testo

For each function, find (a) ƒ(2) and (b) ƒ(-1). See Example 7. ƒ = {(2, 5), (3, 9), (-1, 11), (5, 3)}

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Domanda del libro di testo

Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √4⁄50