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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 63
Capitolo 1, Problema 63

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.


h(x)=−4x2−4x+12h(x)=-4x^{^2}-4x+12

Guida verificata passo dopo passo
1
Identify the original function: The given function is a quadratic function, which can be related to the basic form \( f(x) = x^2 \).
Rewrite the function in vertex form: Start by completing the square for the expression \(-4x^2 - 4x + 12\).
Factor out the coefficient of \(x^2\) from the first two terms: \(-4(x^2 + x) + 12\).
Complete the square inside the parentheses: Add and subtract \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\) inside the parentheses, adjusting for the factor of \(-4\).
Rewrite the function in vertex form: \(h(x) = -4((x + \frac{1}{2})^2 - \frac{1}{4}) + 12\), and simplify to identify shifts and scalings.

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Graph Transformations

Graph transformations involve shifting and scaling functions to create new graphs from original ones. Shifts can be vertical or horizontal, moving the graph up, down, left, or right, while scalings stretch or compress the graph vertically or horizontally. Understanding these transformations helps in visualizing how changes to the function's equation affect its graph.
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Percorso guidato
5:25
Intro to Transformations

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Recognizing the standard form of a quadratic function is essential for identifying its vertex, axis of symmetry, and intercepts.
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6:04
Introduction to Polynomial Functions

Graphing Utilities

Graphing utilities are software or tools that allow users to visualize mathematical functions and their transformations. These tools can plot graphs accurately and provide immediate feedback on the effects of shifts and scalings. Using a graphing utility is a practical way to verify the transformations applied to a function and to explore its behavior visually.
Video consigliato:
Percorso guidato
06:15
Graphing The Derivative
Pratica correlata
Domanda del libro di testo

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.


a. Is this function one-to-one on the interval 0 ≤ t ≤ 4?

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Domanda del libro di testo

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.


b. Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = ƒ⁻¹ (h)

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Domanda del libro di testo

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 4x-3

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Domanda del libro di testo

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 10

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Domanda del libro di testo

Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.


(g o ƒ ) (x) = x²⸍³ + 3

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Domanda del libro di testo

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.

g(x)=−3x2g(x)=-3x^2

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