/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Use the vertex and intercepts to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 17

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)=(x-4)^{2}-1$$

Short Answer

Expert verified
The vertex is at (4, -1). The intercepts are at (5,0), (3,0), and (0,-15). The axis of symmetry is \(x=4\). The domain of \(f(x)\) is \(x \in (-\infty , +\infty)\) and the range is \(y \in [-1, +\infty)\).

Step by step solution

01

Finding the Vertex

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 the function \(f(x)=(x-4)^{2}-1\), the vertex is at the point (4, -1).
02

Finding the Intercepts

To find the x-intercepts, set \(y = f(x)\) to zero and solve for x. \nFor \(f(x)=(x-4)^{2}-1=0\), solving gives \(x = 3\), and \(x = 5\).For the y-intercept, set \(x=0\) in the function \(f(x)\). Evaluating this gives \(y = -15\). So, the intercepts are at (5,0), (3,0) and (0,-15).
03

Finding the Axis of Symmetry

The axis of symmetry for a parabola in vertex form is \(x=h\). Hence, for this function, the axis of symmetry is \(x=4\).
04

Determining the Domain and Range

The domain of a quadratic function is all real numbers since a parabola extends indefinitely to the left and right. So the domain of \(f(x)\) is \(x \in (-\infty , +\infty)\). The range of a quadratic function depends on whether it opens upward or downward. Since \(f(x)\) opens upward, the minimum value it achieves is the y-coordinate of the vertex, which is -1. Therefore, the range of \(f(x)\) is \(y \in [-1, +\infty)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the four-step procedure for solving variation problems given on page 424 to solve. \(C\) varies jointly as \(A\) and \(T . C=175\) when \(A=2100\) and \(T=4 .\) Find \(C\) when \(A=2400\) and \(T=6\).

The functions $$f(x)=0.0875 x^{2}-0.4 x+66.6$$ and $$g(x)=0.0875 x^{2}+1.9 x+11.6$$ model a car's stopping distance, \(f(x)\) or \(g(x),\) in feet, traveling at \(x\) miles per hour. Function \(f\) models stopping distance on dry pavement and function g models stopping distance on wet pavement. The graphs of these functions are shown for \(\\{x | x \geq 30\\} .\) Notice that the figure does not specify which graph is the model for dry roads and which is the model for wet roads. Use this information to solve. (GRAPH CANNOT COPY). a. Use the given functions to find the stopping distance on dry pavement and the stopping distance on wet pavement for a car traveling at 55 miles per hour. Round to the nearest foot. b. Based on your answers to part (a), which rectangular coordinate graph shows stopping distances on dry pavement and which shows stopping distances on wet pavement? c. How well do your answers to part (a) model the actual stopping distances shown in Figure 3.43 on page \(411 ?\) d. Determine speeds on wet pavement requiring stopping distances that exceed the length of one and one-half football fields, or 540 feet. Round to the nearest mile per hour. How is this shown on the appropriate graph of the models?

Sound intensity varies inversely as the square of the distance from the sound source. If you are in a movie theater and you change your seat to one that is twice as far from the speakers, how does the new sound intensity compare to that of your original seat?

Begin by deciding on a product that interests the group because you are now in charge of advertising this product. Members were told that the demand for the product varies directly as the amount spent on advertising and inversely as the price of the product. However, as more money is spent on advertising, the price of your product rises. Under what conditions would members recommend an increased expense in advertising? Once you've determined what your product is, write formulas for the given conditions and experiment with hypothetical numbers. What other factors might you take into consideration in terms of your recommendation? How do these factors affect the demand for your product?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Asproduction level increases, the average cost for a company to produce each unit of its product also increases.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.