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laiz [17]
3 years ago
13

Using a chart of values, graph the function f(x)=3^x

Mathematics
2 answers:
Citrus2011 [14]3 years ago
8 0
Hey there!

For this problem, I would assume that you must make your own chart of values. You can do this by simply plugging in numbers like 1, 2, 3, etc... Since your graph won't be linear, you might want to do some negative numbers, as well. 

Also because your graph isn't linear, you might want to figure out or determine on your own just how far your curve will go. You can then figure out how many times you need to plug in an additional numbers for your function to then graph. 

I've attached a digital graph of this function, perhaps you can find it useful. 

Hope this helped you out! :-)

Ivan3 years ago
4 0
Let's approach this by understanding how an exponential works, to begin.

Any exponential in the form: f(x) = 3^{x} will cut the y-axis once and once only. This is because the exponential graph is a monotonic increasing graph for all real values of x.

We can prove this by taking the derivative. When we take the derivative of a function, we are finding the slope or gradient of the function at an arbitrary point on the graph.

\text{Differentiating the function:} f'(x) = \frac{d(3^{x})}{dx}
f(x) = e^{ln(3^{x})}
f(x) = e^{xln3}
f'(x) = ln(3) \cdot e^{xln3}

Thus, we can conclude for any value of x, the gradient will always be increasing. Now, by plotting some critical values, we can find that the graph hits the y-axis once and once only at the value y = 1.

At this point, we've sort of have an idea as to what the graph will do, but the most critical points are the endpoints, when x approaches infinity and negative infinity. In order to understand what they do at those points, instead of plotting those points, let's use a limit.

\lim_{x \to \pm \infty}(3^{x})

Limits are a crucial tool in understanding what a graph does as the terms get arbitrarily huge and infinitesimally small. We can see that as x approaches infinity, we approach some really big number faster than a  quadratic, linear line, and any function other than factorials.

However, let's approach the negative side. If we were to plug in negative infinity into the function, we get:
f(-\infty) = 3^{-\infty}
f(-\infty) = \frac{1}{3^{\infty}}

Now, we know that for a really big number in the denominator, the function will shrink smaller and smaller to a value, 0. Because the numerator is positive, we can see that the graph will NEVER hit below 0. So, we can conclude that the graph has a limit at x ≠ 0.

Since we know the endpoints, the gradient at any value, and the y-intercept, we can graph the function. Refer to the other answer for the correct graph of 3ˣ.
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raph the equation with a diameter that has endpoints at (-3, 4) and (5, -2). Label the center and at least four points on the ci
andreyandreev [35.5K]

Answer:

Equation:

{x}^{2}   +  {y}^{2} +  2x  - 2y   -  35= 0

The point (0,-5), (0,7), (5,0) and (-7,0)also lie on this circle.

Step-by-step explanation:

We want to find the equation of a circle with a diamterhat hs endpoints at (-3, 4) and (5, -2).

The center of this circle is the midpoint of (-3, 4) and (5, -2).

We use the midpoint formula:

( \frac{x_1+x_2}{2}, \frac{y_1+y_2,}{2} )

Plug in the points to get:

( \frac{ - 3+5}{2}, \frac{ - 2+4}{2} )

( \frac{ -2}{2}, \frac{ 2}{2} )

(  - 1, 1)

We find the radius of the circle using the center (-1,1) and the point (5,-2) on the circle using the distance formula:

r =  \sqrt{ {(x_2-x_1)}^{2} + {(y_2-y_1)}^{2} }

r =  \sqrt{ {(5 -  - 1)}^{2} + {( - 2- - 1)}^{2} }

r =  \sqrt{ {(6)}^{2} + {( - 1)}^{2} }

r =  \sqrt{ 36+ 1 }  =  \sqrt{37}

The equation of the circle is given by:

(x-h)^2 + (y-k)^2 =  {r}^{2}

Where (h,k)=(-1,1) and r=√37 is the radius

We plug in the values to get:

(x- - 1)^2 + (y-1)^2 =  {( \sqrt{37}) }^{2}

(x + 1)^2 + (y - 1)^2 = 37

We expand to get:

{x}^{2}  + 2x  + 1 +  {y}^{2}  - 2y + 1 = 37

{x}^{2}   +  {y}^{2} +  2x  - 2y +2 - 37= 0

{x}^{2}   +  {y}^{2} +  2x  - 2y   -  35= 0

We want to find at least four points on this circle.

We can choose any point for x and solve for y or vice-versa

When y=0,

{x}^{2}   +  {0}^{2} +  2x  - 2(0)  -   35= 0

{x}^{2}   +2x   -   35= 0

(x - 5)(x + 7) = 0

x = 5 \: or \: x =  - 7

The point (5,0) and (-7,0) lies on the circle.

When x=0

{0}^{2}   +  {y}^{2} +  2(0)  - 2y   -  35= 0

{y}^{2} - 2y   -  35= 0

(y - 7)(y + 5) = 0

y = 7 \: or \: y =  - 5

The point (0,-5) and (0,7) lie on this circle.

3 0
3 years ago
Can someone please help me with math. Pls don't take my points.
svlad2 [7]

Answer:

0.09

0.16

0.35

0.65

0.19

0.56

Step-by-step explanation:

3 0
2 years ago
Triangle ABC has vertices at A(–2, 3), B(–3, –6), and C(2, –1). Is triangle ABC a right triangle? If so, which angle is the righ
Len [333]

Answer:

∠ACB is a right angle

Graph:

4 0
3 years ago
Find two solutions to y=4x+3 use x=0 and x=4
Crazy boy [7]
(for x=0) 3 & (for x=4) 19
4 0
3 years ago
Read 2 more answers
Ben’s driving test had 40 questions and he correctly answered 12 more than he missed. How many questions did he miss?
k0ka [10]

Answer:

He missed 14 questions

Step-by-step explanation:

Total = Right + Miss ⇒ t = r + m

Right = Miss + 12 ⇒ r = m + 12

t = 40

Since t = r + m and r = m + 12 then:

t = m + 12 + m

t = 2m + 12

40 = 2m + 12

28 = 2m

m = 14

3 0
3 years ago
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