Math / Algebra

Question $f(x) = c - x$
What do all members of the family of linear functions $f(x) = c - x$ have in common?
All members of the family of linear functions $f(x) = c - x$ have graphs that are lines with slope \_\_\_\_\_\_\_\_\_\_\_\_ and y-intercept \_\_\_\_\_\_\_\_\_\_\_\_.
Sketch several members of the family.
$c = 2$ $c = 1$ $c = 0$ $c = -1$ $c = -2$
$c = 2$ $c = 1$ $c = 0$ $c = -1$ $c = -2$
$c = 2$ $c = 1$ $c = 0$ $c = -1$ $c = -2$
$c = -2$ $c = -1$ $c = 0$ $c = 1$ $c = 2$

Studdy Solution

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!

STEP 11

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

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Studdy Solution

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!

STEP 11

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

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Studdy Solution

STEP 1

What is this asking? What features do the graphs of the functions f(x)=c−xf(x) = c - xf(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 function2. Identify the slope and y-intercept3. Sketch the graphs

STEP 3

Let's **rewrite** our function f(x)=c−xf(x) = c - xf(x)=c−x in the familiar slope-intercept form, f(x)=mx+bf(x) = mx + bf(x)=mx+b.Remember, the **slope** is represented by mmm, and the **y-intercept** is represented by bbb.

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

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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!

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

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 $b$ .

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

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.

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.

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 $(0, 2)$ .

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, $(0, 0)$ .

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 $(0, -1)$ .

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 $c$ .