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Chapter 3: Problem 63

The table represents a linear function. (a) What is \(f(2) ?\) (b) If \(f(x)=-2.5,\) what is the value of \(x ?\) (c) What is the slope of the line? (d) What is the \(y\) -intercept of the line? (e) Using your answers from parts (c) and (d), write an equation for \(f(x)\) $$ \begin{array}{c|c} x & y=f(x) \\ \hline 0 & 3.5 \\ \hline 1 & 2.3 \\ \hline 2 & 1.1 \\ \hline 3 & -0.1 \\ \hline 4 & -1.3 \\ \hline 5 & -2.5 \end{array} $$

Short Answer

Expert verified
f(2) = 1.1; If f(x) = -2.5, then x = 5; Slope = -1.2; y-intercept = 3.5; Equation: f(x) = -1.2x + 3.5

Step by step solution

01

Finding f(2)

Locate the row where x = 2 in the table. The corresponding y value in this row is f(2). From the table, f(2) is 1.1.
02

Solving for x when f(x) = -2.5

Locate the row where the y value is -2.5. The corresponding x value in this row is the solution. From the table, this x value is 5.
03

Determine the slope of the line

The slope (m) of a linear function can be calculated using any two points (x1,y1) and (x2,y2). Here, use the points (0, 3.5) and (1, 2.3). Calculate the slope using the formula \[m = \frac{y2 - y1}{x2 - x1}\] Substituting the values: \[m = \frac{2.3 - 3.5}{1 - 0} = \frac{-1.2}{1} = -1.2\]
04

Finding the y-intercept

The y-intercept is the y value when x is 0. From the table, when x = 0, y = 3.5. Thus, the y-intercept is 3.5.
05

Write the equation of the line

Using the slope from Step 3 and the y-intercept from Step 4, write the linear equation in the form y = mx + b. Substituting the slope (m) = -1.2 and the y-intercept (b) = 3.5, the equation is: \[f(x) = -1.2x + 3.5\]

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Key Concepts

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

Slope Calculation
The slope of a linear function is crucial because it tells us how steep the line is. Mathematically, the slope is the rate at which the y-value changes relative to the x-value. This is represented by the variable 'm' in the linear equation formula. To calculate the slope, we use the standard formula:
  1. > Define two points
    2. (x1, y1) and (x2, y2) from the table.
  2. Apply the formula:


    1. m = \( \frac{y2 - y1}{x2 - x1} \)

    >
Let鈥檚 use the points (0, 3.5) and (1, 2.3):

> > m = \(\frac{2.3 - 3.5}{1 - 0} = \frac{-1.2}{1} = -1.2\)
>> The slope here is -1.2. This means for every unit increase in x, the y-value decreases by 1.2 units.
Y-Intercept
The y-intercept is a fundamental part of a linear equation as it indicates where the line crosses the y-axis. It is represented by 'b' in the linear equation y = mx + b. In simpler terms, it's the value of y when x is 0. Looking at the provided table, we see that when x=0, y=3.5.
Thus, the y-intercept b is 3.5. To visualize this, imagine the graph of the line crossing the y-axis at this point. The y-intercept helps in forming the entire equation of the line, which allows for predicting y-values for other x-values.
Linear Equation
A linear equation is an algebraic expression that models a straight-line graph. The general form is given by y = mx + b, where 'm' is the slope and 'b' is the y-intercept. From our previous calculations, we have:
  • Slope (m) = -1.2
  • Y-intercept (b) = 3.5


Substituting these values into the linear equation formula, we get:
> f(x) = -1.2x + 3.5


This equation now allows us to calculate the y-value for any given x. For example, if x=2, substituting into the linear equation gives:

> f(2) = -1.2(2) + 3.5 = -2.4 + 3.5 = 1.1

This further confirms our earlier finding from the table that f(2) = 1.1.

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Most popular questions from this chapter

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. A cell phone plan includes 900 anytime minutes for \(\$ 60\) per month, plus a one-time activation fee of \(\$ 36 . \mathrm{A}\) Nokia 6650 cell phone is included at no additional charge. (Source: AT\&T.) Let \(x\) represent the number of months of service and \(y\) represent the cost. If you sign a 1-yr contract, how much will this cell phone plan cost? (Assume that you never use more than the allotted number of minutes.)

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ (7,6) \text { and }(7,-8) $$

Solve each problem. The upper deck at U.S. Cellular Field in Chicago has produced, among other complaints, displeasure with its steepness. It is 160 ft from home plate to the front of the upper deck and 250 ft from home plate to the back. The top of the upper deck is 63 ft above the bottom. What is its slope? (Consider the slope as a positive number here.)

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ y-3 x^{2}=2 $$

Find an equation of the line that satisfies the given conditions. (a) Write the equation in slope-intercept form. (b) Write the equation in standard form. Through \((-1,3) ;\) parallel to \(-x+3 y=12\)
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