/*! 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 2 Find the \(y\)-coordinate corres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 2

Find the \(y\)-coordinate corresponding to each \(x\)-coordinate if the functions are \(f(x)=-2 x-5\) and \(g(x)=3.75(2.5)^{x}\). Check your answers with your calculator. a. \(f(6)\) (a) b. \(f(0)\) c. \(g(2)\) d. \(g(-2)\)

Short Answer

Expert verified
For (a) -17, (b) -5, (c) 23.4375, and (d) 0.6.

Step by step solution

01

Understand the Function f(x)

The function \(f(x) = -2x - 5\) is a linear function represented by the equation of a straight line. To find \(f(a)\), substitute \(a\) for \(x\) in the equation and evaluate.
02

Calculate f(6)

Substitute \(x = 6\) into the function \(f(x) = -2x - 5\):\[f(6) = -2(6) - 5\]Calculate:\[f(6) = -12 - 5 = -17\]So, the \(y\)-coordinate for \(x = 6\) is -17.
03

Calculate f(0)

Substitute \(x = 0\) into the function \(f(x) = -2x - 5\):\[f(0) = -2(0) - 5\]Calculate:\[f(0) = 0 - 5 = -5\]So, the \(y\)-coordinate for \(x = 0\) is -5.
04

Understand the Function g(x)

The function \(g(x) = 3.75(2.5)^x\) is an exponential function. To find \(g(a)\), substitute \(a\) for \(x\) and evaluate using exponentiation.
05

Calculate g(2)

Substitute \(x = 2\) into the function \(g(x) = 3.75(2.5)^x\):\[g(2) = 3.75(2.5)^2\]Calculate:\[g(2) = 3.75 imes 6.25 = 23.4375\]So, the \(y\)-coordinate for \(x = 2\) is 23.4375.
06

Calculate g(-2)

Substitute \(x = -2\) into the function \(g(x) = 3.75(2.5)^x\):\[g(-2) = 3.75(2.5)^{-2}\]Calculate using the property \((a^{-b}) = \frac{1}{a^b}\):\[g(-2) = 3.75 \times \left(\frac{1}{6.25}\right) = 0.6\]So, the \(y\)-coordinate for \(x = -2\) is 0.6.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Functions
Linear functions are mathematical expressions that create a straight line when graphed on the coordinate plane. A typical linear function has the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) tells us how steep the line is, while the y-intercept \( b \) is the point where the line crosses the y-axis. These functions are straightforward as every change in \( x \) results in a proportional change in \( y \).
  • In the function \( f(x) = -2x - 5 \), the slope is \(-2\) and the y-intercept is \(-5\).
  • The negative slope indicates the line decreases as \( x \) increases.
Understanding the components of linear functions helps in easily performing calculations and predicting changes in the \( y \)-coordinates.
Exponential Functions
Exponential functions are characterized by their constant multiplicative growth, which means as \( x \) increases, \( y \) values grow by constant factors rather than being added by constant amounts. A standard exponential function looks like \( g(x) = ab^x \).
  • In \( g(x) = 3.75(2.5)^x \), \( 3.75 \) is the initial value (or y-intercept), and \( 2.5 \) is the base of the exponent, determining the growth rate.
  • If \( b > 1 \), the function represents exponential growth. Conversely, if \( 0 < b < 1 \), it indicates exponential decay.
This nature of exponential functions makes them useful in modeling real-world phenomena, such as population growth or radioactive decay, where quantities grow or shrink rapidly.
Substitution Method
The substitution method is a simple technique used to find specific values in functions. To evaluate a function at a particular \( x \)-coordinate, the substitution method involves replacing \( x \) with the given value and then simplifying the expression.
  • For \( f(6) \) using the function \( f(x) = -2x - 5 \), substitute \( 6 \) for \( x \): \( f(6) = -2(6) - 5 \), simplifying to \( -17 \).
  • Similarly, if you want \( g(2) \) for \( g(x) = 3.75(2.5)^x \), substitute \( 2 \) for \( x \): \( g(2) = 3.75(2.5)^2 \), simplifying to \( 23.4375 \).
Whether dealing with linear or exponential functions, substitution helps in directly evaluating the expression's output at specific points.
Y-Coordinate Calculation
Calculating the \( y \)-coordinate is a critical aspect of understanding how functions behave. Each \( y \)-coordinate is derived from substituting the given \( x \)-value into its function.
  • In a linear function like \( f(x) = -2x - 5 \), finding \( y \) for \( x = 0 \) simply gives \( y = -5 \), as the function is linear and straightforward.
  • For exponential functions, such as \( g(x) = 3.75(2.5)^x \), finding \( y \)-values requires careful handling of exponentiation, like calculating \( g(-2) = 0.6 \) by utilizing properties of negative exponents.
This calculation process enhances comprehension of how functions translate mathematical inputs into meaningful outputs, effectively bridging the abstract concept and practical application.

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

If \(x\) represents actual temperature and \(y\) represents wind chill temperature, the equation $$ y=-29+1.4 x $$ approximates the wind chill temperatures for a wind speed of \(40 \mathrm{mi} / \mathrm{h}\). Enter this equation into Y1 on your calculator and find the requested \(x\) - and \(y\)-values. a. What \(x\)-value gives a \(y\)-value of \(-15^{\circ}\) ? Explain how you use the calculator table function to find this answer. b. Enter $$ y=-15 $$ into \(Y_{2}\) on your calculator. Graph both equations. Explain how to use the graph to answer \(15 \mathrm{a}\).

Consider the equation \(y=-12.4-2.5(x+5.4)\). a. Write the equation in intercept form. b. Name the slope and \(y\)-intercept of the equation in \(16 \mathrm{a}\).

Use function notation to write the equation of a line through each pair of points. a. \((0,5)\) and \((1,12)\) (a) b. \((1,5)\) and \((2,12)\)

On graph paper, draw a graph that is a function and has these three properties: \- Domain of \(x\)-values satisfying \(-3 \leq x \leq 5\) \- Range of \(y\)-values satisfying \(-4 \leq y \leq 4\) Includes the points \((-2,3)\) and \((3,-2)\) (a)

Explain why the equation \(x^{2}=-4\) has no solutions.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.