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Ch. R - Algebra Review
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. R - Algebra ReviewProblem 61
Chapter 1, Problem 61

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

Verified step by step guidance
1
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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Key Concepts

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

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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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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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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Domain and Range of Function Transformations
Related Practice
Textbook Question

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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Textbook Question

Add or subtract, as indicated. See Example 4. (3/a - 2) - (1/2 - a)

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Textbook Question

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

Textbook Question

Find each product. See Example 5. (2x + 5)³

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Textbook Question

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

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Textbook Question

Evaluate each expression. See Example 5. -|4.5|