Question

How do you graph `f(x) = 4^x + 2` by plotting points?

Answer 1

Work out some points and plot them.
Choose `x` values and work out `y` values.

Explanation:

You do it in exactly the way that the question asks... plot points.

To find the points which you need to plot, draw a table and choose about `6` values of `x`, some negative, `0` and some positive.

`x: -2" "-1" "0" "1" "2" "3`
Now use each of your `x` values in the given equation and work out the answer. Each answer will be a `y` value:

`x: " "-2" "-1" "0" "1" "2" "3`
`y: " "2.0625" "2.25" "3" "5" "18" "66`

Now draw a set of axes and plot the points. The last one will probably not fit, but you have an idea where it will lie.

Draw the graph by joining the points. You will see that it is an exponential graph. graph{y= 4^x+2 [-7.785, 3.465, 0.405, 6.03]}

Answer 2

Please read the explanation.

Explanation:

`" "`

Exponential Functions have a fixed number as the base and a variable as its exponent.

`color(red)( :. f(x) = 4^x+2` is an exponential function.

Let us also write its Parent Function.

A parent function is the simplest form of the given function.

Let us say `color(blue)(g(x) = 2^x` is the parent function of `color(red)(f(x) = 4^x+2`.

We do this because we can examine the graphs of both the functions to understand the behavior of the given function.

Next, consider the given function.

`color(red)( :. f(x) = 4^x+2`

Create a table of values for this function as shown:

Let us now analyze the graphs of the given function and its parent function:

Observe the following:

(1) `color(red)(f(x)` shifts 2 units up on the y-axis from the graph of `color(blue)(g(x)`, since the value `2` is added as a constant.

(2) If the coefficient of the x-term increases, then the graph leans sharper towards the y-axis.

(3) For any value of `x`, the graph is getting closer and closer to the x-axis but it never touches the x-axis.

(4) For the parent function `color(blue)(g(x)`, we have x-axis as its Horizontal Asymptote.

(5) For the given function `color(red)(f(x)`, we have a Horizontal Asymptote at `color(green)(y=2`.

(6) Exponential growth is bigger and faster than the growth of a polynomial.

(7) As values for `x` grows, then the corresponding y-values increase exponentially.

Hope you find this solution useful.