Find the linear function with the following properties:
[tex]
\begin{array}{c}
f(-4)=-11 \\
f(-10)=3
\end{array}
[/tex]
Answer (2)
Find the slope m using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 3 7 .
Use one of the points and the slope to find the y-intercept b . Substituting ( − 4 , − 11 ) into y = m x + b gives b = − 3 61 .
Write the linear function as f ( x ) = m x + b , which results in f ( x ) = − 3 7 x − 3 61 .
The linear function is f ( x ) = − 3 7 x − 3 61 .
Explanation
Understanding the Problem We are given two points on a linear function and asked to find the equation of the line. The points are ( − 4 , − 11 ) and ( − 10 , 3 ) . A linear function can be written in the form f ( x ) = m x + b , where m is the slope and b is the y-intercept. Our goal is to find the values of m and b .
Calculating the Slope First, we need to find the slope m . The slope is the change in y divided by the change in x . Using the given points, we have: m = x 2 − x 1 y 2 − y 1 = − 10 − ( − 4 ) 3 − ( − 11 ) = − 10 + 4 3 + 11 = − 6 14 = − 3 7 So, the slope m = − 3 7 .
Finding the Y-Intercept Now that we have the slope, we can use one of the given points to find the y-intercept b . Let's use the point ( − 4 , − 11 ) . We plug the values x = − 4 , y = − 11 , and m = − 3 7 into the equation f ( x ) = m x + b :
− 11 = − 3 7 ( − 4 ) + b − 11 = 3 28 + b To solve for b , we subtract 3 28 from both sides: b = − 11 − 3 28 = − 3 33 − 3 28 = − 3 61 So, the y-intercept b = − 3 61 .
Writing the Linear Function Now we have the slope m = − 3 7 and the y-intercept b = − 3 61 . We can write the linear function as: f ( x ) = − 3 7 x − 3 61 Thus, the linear function is f ( x ) = − 3 7 x − 3 61 .
Final Answer The linear function with the given properties is f ( x ) = − 3 7 x − 3 61 .
Examples
Linear functions are incredibly useful in everyday life. For example, if you are saving money at a constant rate, the amount you save over time can be modeled with a linear function. Similarly, if you are driving at a constant speed, the distance you travel over time can be modeled with a linear function. Understanding linear functions helps you predict future outcomes based on current trends.
To find the linear function with the properties f(-4) = -11 and f(-10) = 3, we calculated the slope to be -7/3 and the y-intercept to be -61/3. Therefore, the linear function is f(x) = -7/3 x - 61/3. This provides the equation for the line that connects the two given points.
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