Question

1. the function $f(x) = \frac{1}{2}x + 70$ represents mike’s test grade, $f(x)$, after $x$ minutes of studying. what does the slope represent in this situation?\

\
\bigcirc for every 5 minutes of study time, his grade decreases 10 points.\
\bigcirc for every 10 minutes of study time, his grade increases 5 points.\
\bigcirc for every 10 minutes of study time, his grade decreases 5 points.\
\bigcirc for every 5 minutes of study time, his grade increases 10 points.

Explanation:

Step1: Recall slope-intercept form

The function is in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For \(f(x)=\frac{1}{2}x + 70\), the slope \(m=\frac{1}{2}\).

Step2: Interpret the slope

The slope \(\frac{1}{2}\) means that for every 1 unit increase in \(x\) (study time in minutes), the grade \(f(x)\) increases by \(\frac{1}{2}\) points. Let's find the change in grade for a change in study time.
If \(x\) increases by 10 minutes ( \(\Delta x = 10\) ), then \(\Delta f(x)=m\times\Delta x=\frac{1}{2}\times10 = 5\) points. So for every 10 minutes of study time, his grade increases by 5 points.

Answer:

For every 10 minutes of study time, his grade increases 5 points.