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Chapter 2: Problem 80

Find a function whose graph is the given curve. The line segment joining the points (-3,-2) and (6,3)

Short Answer

Expert verified
The function is \( f(x) = \frac{5}{9}x - \frac{1}{3} \).

Step by step solution

01

Understanding the Problem

We are asked to find an equation of a line that passes through the points (-3, -2) and (6, 3).
02

Finding the Slope

The slope of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. For our points, \((x_1, y_1) = (-3, -2)\) and \((x_2, y_2) = (6, 3)\). Plugging the values into the formula, \[ m = \frac{3 - (-2)}{6 - (-3)} = \frac{3 + 2}{6 + 3} = \frac{5}{9} \].
03

Using Point-Slope Form

Once we have the slope \(m\), we can use the point-slope form of a line's equation, \[ y - y_1 = m(x - x_1) \], which uses any point on the line \((x_1, y_1)\). Using the point \((-3, -2)\) and the slope \(\frac{5}{9}\), the equation becomes \[ y + 2 = \frac{5}{9}(x + 3) \].
04

Simplifying the Equation

We need to rearrange the equation \( y + 2 = \frac{5}{9}(x + 3) \) to the slope-intercept form, \( y = mx + b \). First, distribute the slope \( \frac{5}{9} \): \[ y + 2 = \frac{5}{9}x + \frac{5}{3} \]. Then, subtract 2 from both sides to solve for \(y\): \[ y = \frac{5}{9}x + \frac{5}{3} - 2 \]. Simplify further: \[ y = \frac{5}{9}x + \frac{5}{3} - \frac{6}{3} \]. \[ y = \frac{5}{9}x - \frac{1}{3} \].

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form is essential in algebra. This specific way of writing a line's equation, represented as \( y = mx + b \), is incredibly useful. In this form:
  • \( m \) stands for the slope, which tells us how steep the line is.
  • \( b \) represents the y-intercept, or where the line crosses the y-axis.
To illustrate with our example, we already calculated the equation \( y = \frac{5}{9}x - \frac{1}{3} \). Here, the slope \( m \) is \( \frac{5}{9} \), indicating the line rises by 5 units for every 9 units it runs horizontally. Meanwhile, the y-intercept is \( -\frac{1}{3} \), showing the point where the line touches the y-axis. Converting other line forms to this format can simplify understanding and graphing the line.
Point-Slope Form
Before reaching the slope-intercept form, we often use the point-slope form. This form is especially handy when you know a point on the line and the slope. The point-slope form equation is \( y - y_1 = m(x - x_1) \). It uses:
  • \((x_1, y_1)\) which is a known point on the line.
  • \(m\) which is the slope.
In our case, we used the point \((-3, -2)\) with the slope \( \frac{5}{9} \). So our equation is \( y + 2 = \frac{5}{9}(x + 3) \). With this equation, we are indicating the line passes through the given point with the calculated slope. By rearranging and solving this form, you can reach the slope-intercept form.
Calculating Slope
The slope of a line is crucial as it defines how the line angles on a graph. To find the slope when two points on a line are given, you use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  • \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points.
  • The numerator \((y_2 - y_1)\) indicates the vertical change or rise.
  • The denominator \((x_2 - x_1)\) represents the horizontal change or run.
For the points \((-3, -2)\) and \((6, 3)\), we plug these values into our formula: \( m = \frac{3 - (-2)}{6 - (-3)} = \frac{5}{9} \). This shows the line rises 5 units for every 9 units it moves to the right. Understanding this step helps reveal the steepness and direction of any line.

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Most popular questions from this chapter

Find a function whose graph is the given curve. The bottom half of the circle \(x^{2}+y^{2}=9\)

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. \(f(x)=x^{c}\) (a) \(c=\frac{1}{2}, \frac{1}{4}, \frac{1}{6} ; \quad[-1,4]\) by \([-1,3]\) (b) \(c=1, \frac{1}{3}, \frac{1}{5} ; \quad[-3,3]\) by \([-2,2]\) (c) How does the value of \(c\) affect the graph?

Sums of Even and Odd Functions If \(f\) and \(g\) are both even functions, is \(f+g\) necessarily even? If both are odd, is their sum necessarily odd? What can you say about the sum if one is odd and one is even? In each case, prove your answer.

Express the function in the form \(f \circ g\) $$G(x)=\frac{x^{2}}{x^{2}+4}$$

The graphs of \(f(x)=x^{2}-4\) and \(g(x)=\left|x^{2}-4\right|\) are shown. Explain how the graph of \(g\) is obtained from the graph of \(f\) (Graph can't copy)
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