Math Snap

STEP 1

What is this asking?
What features do the graphs of the functions $f(x) = c - x$ share, and what do they look like?
Watch out!
Don't mix up the slope and the y-intercept!

STEP 2

1. Rewrite the function
2. Identify the slope and y-intercept
3. Sketch the graphs

STEP 3

Let's rewrite our function $f(x) = c - x$ in the familiar slope-intercept form, $f(x) = mx + b$.
Remember, the slope is represented by $m$, and the y-intercept is represented by $b$.

STEP 4

We can rewrite $f(x) = c - x$ as $f(x) = -x + c$.
See how we just switched the order?
Addition is commutative, meaning $a + b = b + a$.

STEP 5

Now, compare $f(x) = -x + c$ with $f(x) = mx + b$.
Notice that the coefficient of $x$ in our rewritten function is $-1$.
This tells us that the slope, $m$, is $-1$.
Every function in this family will have this same slope.

STEP 6

The constant term in our rewritten function is $c$.
This means the y-intercept, $b$, is $c$.
Since $c$ can change, the y-intercept changes, giving us different lines in the family.

STEP 7

Let's sketch a few examples!
If $c = 2$, our function is $f(x) = -x + 2$.
This line has a slope of $-1$ and crosses the y-axis at $(0, 2)$.

STEP 8

If $c = 0$, our function is $f(x) = -x + 0$, which simplifies to $f(x) = -x$.
The slope is still $-1$, and it crosses the y-axis at the origin, $(0, 0)$.

STEP 9

If $c = -1$, our function is $f(x) = -x - 1$.
Again, the slope is $-1$, and this time, the line crosses the y-axis at $(0, -1)$.

STEP 10

Notice how all the lines we sketched are parallel because they all have the same slope of $-1$.
They just shift up or down depending on the value of $c$, which is the y-intercept!

All members of the family of linear functions $f(x) = c - x$ have graphs that are lines with slope $-1$ and y-intercept $c$.