Suppose f is a linear function with slope 2 and such that f(-3)=8. Find f(x).
Understand the Problem
The question asks you to find the equation of a linear function f(x). You are given that the slope of the function is 2, and that f(-3) = 8. You need to use this information to determine the y-intercept and thus find the correct equation for f(x) from the options provided.
Answer
$f(x) = 2x + 14$
Answer for screen readers
$f(x) = 2x + 14$
Steps to Solve
- Write the general form of a linear function
A linear function can be expressed as $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
- Substitute the given slope
We are given that the slope $m = 2$. Substitute this into the equation to get $f(x) = 2x + b$.
- Use the given point to find the y-intercept
We are given that $f(-3) = 8$. Substitute $x = -3$ into the equation $f(x) = 2x + b$ to solve for $b$: $8 = 2(-3) + b$
- Solve for b
Simplify the equation to solve for $b$:
$8 = -6 + b$
$b = 8 + 6$
$b = 14$
- Write the final equation
Substitute the value of $b$ back into the equation $f(x) = 2x + b$ to get the final equation: $f(x) = 2x + 14$
$f(x) = 2x + 14$
More Information
The equation $f(x) = 2x + 14$ represents a line with a slope of 2 and a y-intercept of 14. When $x = -3$, $f(x) = 2(-3) + 14 = -6 + 14 = 8$, which satisfies the given condition $f(-3) = 8$.
Tips
A common mistake is to incorrectly substitute the x-value when solving for the y-intercept ($b$). Another mistake is to confuse the slope and y-intercept, inserting them in the wrong places in the equation $f(x) = mx + b$. Also, there might be mistakes with the arithmetic, leading to incorrect calculation of $b$.
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