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Chapter 2: Problem 120
Suppose that a function \(g\) is such that \(g(-1)=-7\) and \(g(3)=8 .\) Find aformula for \(g\) if \(g(x)\) is of the form \(g(x)=m x+b,\) where \(m\) and \(b\) areconstants.
Short Answer
Expert verified
The formula for \(g(x)\) is \(g(x) = 3.75x - 3.25\).
Step by step solution
01
- Understand the Given Information
We are given two points on the line defined by the function: 1. \(\left( -1, -7 \right)\)2. \(\left( 3, 8 \right)\)These points will help to find the slope \(m\) and the y-intercept \(b\).
02
- Find the Slope (m)
The slope \(m\) of a line passing through two points \( (x_1, y_1) \) and \((x_2, y_2)\) is calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute \( (x_1, y_1) = (-1, -7)\) and \( (x_2, y_2) = ()3, 8)\) to get:\[m = \frac{8 - (-7)}{3 - (-1)} = \frac{15}{4} = 3.75\]
03
- Use One Point to Solve for y-intercept (b)
Therefore, \(b = -3.25\).
04
- Write the Equation
Now that we have \(m\) and \(b\), substitute these values back into the equation \g(x) = mx + b\ to get the final formula.\[g(x) = 3.75x - 3.25\]
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope-intercept form
To describe the equation of a straight line, we often use the slope-intercept form. The general formula is: \[ g(x) = mx + b \]Here, \(m\) represents the slope of the line. The slope tells us how steep the line is. The \(b\) represents the y-intercept. This is where the line crosses the y-axis.
Understanding the slope-intercept form is essential. It helps us graph linear equations easily and understand their behavior. When you know \(m\) and \(b\), you have all the information needed to draw the line.
finding slope
The slope \(m\) measures the 'rise' over the 'run' between two points. To find the slope between points \((x_1, y_1)\) and \((x_2, y_2)\), use:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For the given points \((-1, -7)\) and \( (3, 8)\):
\[ m = \frac{8 - ( -7 )}{3 - (-1)} = \frac{15}{4} = 3.75 \]This calculation shows that for every 4 units to the right (run), the line goes up by 15 units (rise).
Remember:
- 'Rise': Change in y-values
- 'Run': Change in x-values
y-intercept calculation
After finding the slope, we need to determine the y-intercept, \(b\). The y-intercept is the point where the line crosses the y-axis. To find \ b, \ use one of the points and the slope.
The formula of the line is:
\[ g(x) = mx + b \] Substitute in the values of one of the points. Let's use \( (3, 8) \). We have:
\[ 8 = 3.75(3) + b \]Solve for \ b:\
\[ 8 = 11.25 + b \]\[ b = 8 - 11.25 = -3.25 \]Therefore, the y-intercept is \b = -3.25\.
linear equation derivation
Finally, we combine the slope and y-intercept to write the final linear equation. We already know:
- The slope \(m = 3.75\)
- The y-intercept \(b = -3.25\)
Substitute these into the slope-intercept form:
\[ g(x) = 3.75x - 3.25 \]This is the equation of the line that passes through the given points \((-1, -7)\) and \((3, 8)\).
Understanding this process helps in writing equations for any linear functions. Always start with identifying points on the line. Then, find the slope and y-intercept, and finally, assemble them into the complete equation.
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