Math / Algebra

Question(1 point) Given the function $f(x)=-2 \sqrt{8-x}+8$ The $y$ -intercept of the function is $\square$ The $x$ -intercept of the function is $\square$ help (points) After submitting an answer, you can get a hint and click "Show Me Another" to see a simila

Studdy Solution

STEP 1

What is this asking? We need to find the points where this curvy function $f(x)$ crosses the x-axis and the y-axis! Watch out! Remember that the y-intercept happens when $x = 0$ and the x-intercept happens when $y = 0$ , so don't mix them up!

STEP 2

1. Find the y-intercept
2. Find the x-intercept

STEP 3

To find the y-intercept, we need to figure out where the function crosses the y-axis.
This happens when $x = 0$ .
So, let's substitute $x = 0$ into our function $f(x)$ .

STEP 4

$f(0) = -2 \cdot \sqrt{8 - 0} + 8$ $f(0) = -2 \cdot \sqrt{8} + 8$

STEP 5

We can simplify $\sqrt{8}$ by noticing that $8 = 4 \cdot 2$ , so $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$ .
Let's plug that in: $f(0) = -2 \cdot 2\sqrt{2} + 8$ $f(0) = -4\sqrt{2} + 8$

STEP 6

So, our y-intercept is $(0, -4\sqrt{2} + 8)$ !

STEP 7

Now, for the x-intercept!
This is where our function crosses the x-axis, meaning $f(x) = 0$ , or $y = 0$ .
Let's set our function equal to zero and solve for $x$ : $0 = -2\sqrt{8 - x} + 8$

STEP 8

First, let's isolate the square root term.
We can do this by subtracting 8 from both sides: $-8 = -2\sqrt{8 - x}$

STEP 9

Now, let's divide both sides by -2: $\frac{-8}{-2} = \frac{-2\sqrt{8 - x}}{-2}$ $4 = \sqrt{8 - x}$

STEP 10

To get rid of the square root, we can square both sides: $4^2 = (\sqrt{8 - x})^2$ $16 = 8 - x$

STEP 11

Now, we can solve for $x$ .
Let's subtract 8 from both sides: $16 - 8 = 8 - x - 8$ $8 = -x$

STEP 12

Finally, multiply both sides by -1 to get: $-8 = x$ $x = -8$

STEP 13

So, our x-intercept is $(-8, 0)$ !

STEP 14

The y-intercept is $(0, -4\sqrt{2} + 8)$ . The x-intercept is $(-8, 0)$ .

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Question(1 point) Given the function $f(x)=-2 \sqrt{8-x}+8$ The $y$ -intercept of the function is $\square$ The $x$ -intercept of the function is $\square$ help (points) After submitting an answer, you can get a hint and click "Show Me Another" to see a simila

Studdy Solution

STEP 1

What is this asking? We need to find the points where this curvy function $f(x)$ crosses the x-axis and the y-axis! Watch out! Remember that the y-intercept happens when $x = 0$ and the x-intercept happens when $y = 0$ , so don't mix them up!

STEP 2

1. Find the y-intercept
2. Find the x-intercept

STEP 3

To find the y-intercept, we need to figure out where the function crosses the y-axis.
This happens when $x = 0$ .
So, let's substitute $x = 0$ into our function $f(x)$ .

STEP 4

$f(0) = -2 \cdot \sqrt{8 - 0} + 8$ $f(0) = -2 \cdot \sqrt{8} + 8$

STEP 5

We can simplify $\sqrt{8}$ by noticing that $8 = 4 \cdot 2$ , so $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$ .
Let's plug that in: $f(0) = -2 \cdot 2\sqrt{2} + 8$ $f(0) = -4\sqrt{2} + 8$

STEP 6

So, our y-intercept is $(0, -4\sqrt{2} + 8)$ !

STEP 7

Now, for the x-intercept!
This is where our function crosses the x-axis, meaning $f(x) = 0$ , or $y = 0$ .
Let's set our function equal to zero and solve for $x$ : $0 = -2\sqrt{8 - x} + 8$

STEP 8

First, let's isolate the square root term.
We can do this by subtracting 8 from both sides: $-8 = -2\sqrt{8 - x}$

STEP 9

Now, let's divide both sides by -2: $\frac{-8}{-2} = \frac{-2\sqrt{8 - x}}{-2}$ $4 = \sqrt{8 - x}$

STEP 10

To get rid of the square root, we can square both sides: $4^2 = (\sqrt{8 - x})^2$ $16 = 8 - x$

STEP 11

Now, we can solve for $x$ .
Let's subtract 8 from both sides: $16 - 8 = 8 - x - 8$ $8 = -x$

STEP 12

Finally, multiply both sides by -1 to get: $-8 = x$ $x = -8$

STEP 13

So, our x-intercept is $(-8, 0)$ !

STEP 14

The y-intercept is $(0, -4\sqrt{2} + 8)$ . The x-intercept is $(-8, 0)$ .

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Question(1 point) Given the function f(x)=−28−x+8f(x)=-2 \sqrt{8-x}+8f(x)=−28−x​+8 The yyy-intercept of the function is □\square□ The xxx-intercept of the function is □\square□ help (points) After submitting an answer, you can get a hint and click "Show Me Another" to see a simila

Studdy Solution

STEP 1

What is this asking? We need to find the points where this curvy function f(x)f(x)f(x) crosses the x-axis and the y-axis! Watch out! Remember that the y-intercept happens when x=0x = 0x=0 and the x-intercept happens when y=0y = 0y=0, so don't mix them up!

STEP 2

1. Find the y-intercept2. Find the x-intercept

STEP 3

To find the y-intercept, we need to figure out where the function crosses the y-axis.This happens when x=0x = 0x=0.So, let's **substitute** x=0x = 0x=0 into our function f(x)f(x)f(x).

STEP 4

f(0)=−2⋅8−0+8 f(0) = -2 \cdot \sqrt{8 - 0} + 8 f(0)=−2⋅8−0​+8 f(0)=−2⋅8+8 f(0) = -2 \cdot \sqrt{8} + 8 f(0)=−2⋅8​+8

STEP 5

STEP 6

So, our **y-intercept** is (0,−42+8)(0, -4\sqrt{2} + 8)(0,−42​+8)!

STEP 7

Now, for the x-intercept!This is where our function crosses the x-axis, meaning f(x)=0f(x) = 0f(x)=0, or y=0y = 0y=0.Let's **set** our function equal to zero and **solve for** xxx: 0=−28−x+8 0 = -2\sqrt{8 - x} + 8 0=−28−x​+8

STEP 8

First, let's **isolate** the square root term.We can do this by **subtracting** 8 from both sides: −8=−28−x -8 = -2\sqrt{8 - x} −8=−28−x​

STEP 9

Now, let's **divide** both sides by -2: −8−2=−28−x−2 \frac{-8}{-2} = \frac{-2\sqrt{8 - x}}{-2} −2−8​=−2−28−x​​ 4=8−x 4 = \sqrt{8 - x} 4=8−x​

STEP 10

To get rid of the square root, we can **square** both sides: 42=(8−x)2 4^2 = (\sqrt{8 - x})^2 42=(8−x​)2 16=8−x 16 = 8 - x 16=8−x

STEP 11

Now, we can **solve for** xxx.Let's **subtract** 8 from both sides: 16−8=8−x−8 16 - 8 = 8 - x - 8 16−8=8−x−8 8=−x 8 = -x 8=−x

STEP 12

Finally, **multiply** both sides by -1 to get: −8=x -8 = x −8=x x=−8 x = -8 x=−8

STEP 13

So, our **x-intercept** is (−8,0)(-8, 0)(−8,0)!

STEP 14

The y-intercept is (0,−42+8)(0, -4\sqrt{2} + 8)(0,−42​+8). The x-intercept is (−8,0)(-8, 0)(−8,0).

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Question(1 point) Given the function $f(x)=-2 \sqrt{8-x}+8$ The $y$ -intercept of the function is $\square$ The $x$ -intercept of the function is $\square$ help (points) After submitting an answer, you can get a hint and click "Show Me Another" to see a simila

What is this asking? We need to find the points where this curvy function $f(x)$ crosses the x-axis and the y-axis! Watch out! Remember that the y-intercept happens when $x = 0$ and the x-intercept happens when $y = 0$ , so don't mix them up!

1. Find the y-intercept
2. Find the x-intercept

To find the y-intercept, we need to figure out where the function crosses the y-axis.
This happens when $x = 0$ .
So, let's substitute $x = 0$ into our function $f(x)$ .

$f(0) = -2 \cdot \sqrt{8 - 0} + 8$ $f(0) = -2 \cdot \sqrt{8} + 8$

So, our y-intercept is $(0, -4\sqrt{2} + 8)$ !

Now, for the x-intercept!
This is where our function crosses the x-axis, meaning $f(x) = 0$ , or $y = 0$ .
Let's set our function equal to zero and solve for $x$ : $0 = -2\sqrt{8 - x} + 8$

First, let's isolate the square root term.
We can do this by subtracting 8 from both sides: $-8 = -2\sqrt{8 - x}$

Now, let's divide both sides by -2: $\frac{-8}{-2} = \frac{-2\sqrt{8 - x}}{-2}$ $4 = \sqrt{8 - x}$

To get rid of the square root, we can square both sides: $4^2 = (\sqrt{8 - x})^2$ $16 = 8 - x$

Now, we can solve for $x$ .
Let's subtract 8 from both sides: $16 - 8 = 8 - x - 8$ $8 = -x$

Finally, multiply both sides by -1 to get: $-8 = x$ $x = -8$

So, our x-intercept is $(-8, 0)$ !

The y-intercept is $(0, -4\sqrt{2} + 8)$ . The x-intercept is $(-8, 0)$ .