Math / Algebra

Question $g(x)$ is a growth $\square$ function. $g(x)=\frac{1}{2}(6)^{x}+7$ $g(x)$ has a $y$ - intercept of $(0$ , $\square$ 7.5 $g(x)$ has a $x$ -intercept of there are no $x$ -intercepts $\square$ , 0). $g(x)$ has an asymptote of $y=7$ $\square$

Studdy Solution

STEP 1

What is this asking? We're given a function $g(x)$ and need to figure out if it's a growth or decay function, find its y-intercept, see if it has an x-intercept, and identify its asymptote! Watch out! Remember, exponential functions can be tricky with their intercepts and asymptotes, so let's be extra careful!

STEP 2

1. Determine Growth or Decay
2. Find the Y-Intercept
3. Find the X-Intercept
4. Identify the Asymptote

STEP 3

Alright, we've got our function $g(x) = \frac{1}{2}(6)^{x} + 7$ .
The key part here is the base of the exponential term, which is 6.

STEP 4

Since 6 is greater than 1, this tells us $g(x)$ is a growth function!
It's going to get bigger and bigger as $x$ increases.
Woohoo!

STEP 5

To find the y-intercept, we need to set $x$ to 0 and see what $g(x)$ becomes.
It's like seeing where our function starts on the y-axis!

STEP 6

$g(0) = \frac{1}{2}(6)^{0} + 7$

STEP 7

Remember, anything raised to the power of 0 is 1, so $6^0 = 1$ .

STEP 8

$g(0) = \frac{1}{2} \cdot 1 + 7$ $g(0) = \frac{1}{2} + 7$ $g(0) = \frac{1}{2} + \frac{14}{2}$ $g(0) = \frac{15}{2}$ $g(0) = 7.5$

STEP 9

So, our y-intercept is $(0, 7.5)$ .
Boom!

STEP 10

Now, for the x-intercept, we set $g(x)$ to 0 and solve for $x$ .
This is where our function crosses the x-axis.

STEP 11

$0 = \frac{1}{2}(6)^{x} + 7$

STEP 12

Let's isolate the exponential term: $-7 = \frac{1}{2}(6)^{x}$ $-14 = (6)^{x}$

STEP 13

Uh oh!
We have a problem.
An exponential term with a positive base (like 6) can never be equal to a negative number (like -14).
This means there's no x-intercept!
Our function will never touch the x-axis.

STEP 14

The asymptote is like a line that our function gets really, really close to but never actually touches.
For exponential functions in the form $g(x) = a \cdot b^x + c$ , the asymptote is $y = c$ .

STEP 15

In our case, $c = 7$ , so our asymptote is $y = 7$ .
That means our function will get closer and closer to the line $y = 7$ as $x$ gets smaller and smaller, but it will never actually reach it!

STEP 16

$g(x)$ is a growth function.
The y-intercept is $(0, 7.5)$ .
There are no x-intercepts.
The asymptote is $y = 7$ .

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Question $g(x)$ is a growth $\square$ function. $g(x)=\frac{1}{2}(6)^{x}+7$ $g(x)$ has a $y$ - intercept of $(0$ , $\square$ 7.5 $g(x)$ has a $x$ -intercept of there are no $x$ -intercepts $\square$ , 0). $g(x)$ has an asymptote of $y=7$ $\square$

Studdy Solution

STEP 1

STEP 2

1. Determine Growth or Decay
2. Find the Y-Intercept
3. Find the X-Intercept
4. Identify the Asymptote

STEP 3

Alright, we've got our function $g(x) = \frac{1}{2}(6)^{x} + 7$ .
The key part here is the base of the exponential term, which is 6.

STEP 4

Since 6 is greater than 1, this tells us $g(x)$ is a growth function!
It's going to get bigger and bigger as $x$ increases.
Woohoo!

STEP 5

To find the y-intercept, we need to set $x$ to 0 and see what $g(x)$ becomes.
It's like seeing where our function starts on the y-axis!

STEP 6

$g(0) = \frac{1}{2}(6)^{0} + 7$

STEP 7

Remember, anything raised to the power of 0 is 1, so $6^0 = 1$ .

STEP 8

$g(0) = \frac{1}{2} \cdot 1 + 7$ $g(0) = \frac{1}{2} + 7$ $g(0) = \frac{1}{2} + \frac{14}{2}$ $g(0) = \frac{15}{2}$ $g(0) = 7.5$

STEP 9

So, our y-intercept is $(0, 7.5)$ .
Boom!

STEP 10

Now, for the x-intercept, we set $g(x)$ to 0 and solve for $x$ .
This is where our function crosses the x-axis.

STEP 11

$0 = \frac{1}{2}(6)^{x} + 7$

STEP 12

Let's isolate the exponential term: $-7 = \frac{1}{2}(6)^{x}$ $-14 = (6)^{x}$

STEP 13

Uh oh!
We have a problem.
An exponential term with a positive base (like 6) can never be equal to a negative number (like -14).
This means there's no x-intercept!
Our function will never touch the x-axis.

STEP 14

The asymptote is like a line that our function gets really, really close to but never actually touches.
For exponential functions in the form $g(x) = a \cdot b^x + c$ , the asymptote is $y = c$ .

STEP 15

In our case, $c = 7$ , so our asymptote is $y = 7$ .
That means our function will get closer and closer to the line $y = 7$ as $x$ gets smaller and smaller, but it will never actually reach it!

STEP 16

$g(x)$ is a growth function.
The y-intercept is $(0, 7.5)$ .
There are no x-intercepts.
The asymptote is $y = 7$ .

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Studdy Solution

STEP 1

STEP 2

1. Determine Growth or Decay2. Find the Y-Intercept3. Find the X-Intercept4. Identify the Asymptote

STEP 3

Alright, we've got our function g(x)=12(6)x+7g(x) = \frac{1}{2}(6)^{x} + 7g(x)=21​(6)x+7.The key part here is the **base** of the exponential term, which is **6**.

STEP 4

Since **6** is greater than **1**, this tells us g(x)g(x)g(x) is a **growth** function!It's going to get bigger and bigger as xxx increases.Woohoo!

STEP 5

To find the y-intercept, we need to set xxx to **0** and see what g(x)g(x)g(x) becomes.It's like seeing where our function starts on the y-axis!

STEP 6

g(0)=12(6)0+7g(0) = \frac{1}{2}(6)^{0} + 7g(0)=21​(6)0+7

STEP 7

Remember, *anything* raised to the power of **0** is **1**, so 60=16^0 = 160=1.

STEP 8

g(0)=12⋅1+7g(0) = \frac{1}{2} \cdot 1 + 7g(0)=21​⋅1+7 g(0)=12+7g(0) = \frac{1}{2} + 7g(0)=21​+7 g(0)=12+142g(0) = \frac{1}{2} + \frac{14}{2}g(0)=21​+214​g(0)=152g(0) = \frac{15}{2}g(0)=215​g(0)=7.5g(0) = 7.5g(0)=7.5

STEP 9

So, our y-intercept is (0,7.5)(0, 7.5)(0,7.5).Boom!

STEP 10

Now, for the x-intercept, we set g(x)g(x)g(x) to **0** and solve for xxx.This is where our function crosses the x-axis.

STEP 11

0=12(6)x+70 = \frac{1}{2}(6)^{x} + 70=21​(6)x+7

STEP 12

Let's isolate the exponential term: −7=12(6)x-7 = \frac{1}{2}(6)^{x}−7=21​(6)x −14=(6)x-14 = (6)^{x}−14=(6)x

STEP 13

Uh oh!We have a problem.An exponential term with a **positive base** (like **6**) can *never* be equal to a **negative number** (like **-14**).This means there's **no x-intercept**!Our function will never touch the x-axis.

STEP 14

The asymptote is like a line that our function gets really, really close to but never actually touches.For exponential functions in the form g(x)=a⋅bx+cg(x) = a \cdot b^x + cg(x)=a⋅bx+c, the asymptote is y=cy = cy=c.

STEP 15

In our case, c=7c = 7c=7, so our asymptote is y=7y = 7y=7.That means our function will get closer and closer to the line y=7y = 7y=7 as xxx gets smaller and smaller, but it will never actually reach it!

STEP 16

g(x)g(x)g(x) is a **growth** function.The y-intercept is (0,7.5)(0, 7.5)(0,7.5).There are **no x-intercepts**.The asymptote is y=7y = 7y=7.

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1. Determine Growth or Decay
2. Find the Y-Intercept
3. Find the X-Intercept
4. Identify the Asymptote

Alright, we've got our function $g(x) = \frac{1}{2}(6)^{x} + 7$ .
The key part here is the base of the exponential term, which is 6.

Since 6 is greater than 1, this tells us $g(x)$ is a growth function!
It's going to get bigger and bigger as $x$ increases.
Woohoo!

To find the y-intercept, we need to set $x$ to 0 and see what $g(x)$ becomes.
It's like seeing where our function starts on the y-axis!

$g(0) = \frac{1}{2}(6)^{0} + 7$

Remember, anything raised to the power of 0 is 1, so $6^0 = 1$ .

$g(0) = \frac{1}{2} \cdot 1 + 7$ $g(0) = \frac{1}{2} + 7$ $g(0) = \frac{1}{2} + \frac{14}{2}$ $g(0) = \frac{15}{2}$ $g(0) = 7.5$

So, our y-intercept is $(0, 7.5)$ .
Boom!

Now, for the x-intercept, we set $g(x)$ to 0 and solve for $x$ .
This is where our function crosses the x-axis.

$0 = \frac{1}{2}(6)^{x} + 7$

Let's isolate the exponential term: $-7 = \frac{1}{2}(6)^{x}$ $-14 = (6)^{x}$

Uh oh!
We have a problem.
An exponential term with a positive base (like 6) can never be equal to a negative number (like -14).
This means there's no x-intercept!
Our function will never touch the x-axis.

The asymptote is like a line that our function gets really, really close to but never actually touches.
For exponential functions in the form $g(x) = a \cdot b^x + c$ , the asymptote is $y = c$ .

In our case, $c = 7$ , so our asymptote is $y = 7$ .
That means our function will get closer and closer to the line $y = 7$ as $x$ gets smaller and smaller, but it will never actually reach it!

$g(x)$ is a growth function.
The y-intercept is $(0, 7.5)$ .
There are no x-intercepts.
The asymptote is $y = 7$ .