Question 622·Medium·Algebra

0 / 700

The table shows four values of $x$ and their corresponding values of $f(x)$ .

$x$	10	15	20	25
$f(x)$	82	137	192	247

There is a linear relationship between $x$ and $f(x)$ that is defined by the equation $f(x)=mx-28$ , where $m$ is a constant. What is the value of $m$ ?

Your answer

In a linear function, the slope $m$ is the change in output for each $1$ increase in input. When the rule gives the output at $x=0$ , one input-output pair is enough to find an unknown slope. Substitute a pair from the table into the rule, then use a Desmos regression to find $m$ . A table output isn't itself the slope.

Hints

Hint 1
The function notation $f(10)$ means the output when the input is $10$ . Match that input to its output in the table. What equation does the first pair give you?
Hint 2
In $f(x)=mx-28$ , $m$ multiplies the input, while $-28$ stays fixed. Put both values from one table pair into the rule before finding the unknown.

Step-by-step

Approach 1: Use one table pair and the given rule
Step 1
Turn a table pair into a function fact
Read the table by matching each input to its output: $10$ is paired with $82$ . Since $f(10)$ means the output when $x=10$ , the table tells you $f(10)=82$ .
Step 2
Put the pair into the given rule
In the given linear rule $f(x)=mx-28$ , replace $x$ with $10$ and $f(10)$ with $82$ :
$82=10m-28.$
Keep the $-28$ . It's the fixed part of the rule, not the value of $m$ .
Step 3
Find the missing coefficient
Type $82\sim10m-28$ in Desmos. The tilde $\sim$ asks Desmos to find the missing constant instead of graphing the equation. Under PARAMETERS, it reports $m=11$ . That's the number multiplying $x$ in the given rule. Grid in 11.

Approach 2: Find the slope from two table pairs
Step 1
Compare matching changes
Use the pairs $(10,82)$ and $(15,137)$ . The slope is the change in output divided by the change in input. Subtract in the same order on top and bottom:
$m=\frac{137-82}{15-10}.$
Step 2
Calculate the rate
Type $\frac{137-82}{15-10}$ in Desmos; it prints $11$ . So the output grows by $11$ for each $1$ increase in $x$ . That rate is the slope, $m=11$ . Grid in 11.

Question 622·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this