y=1/2 times 4 to the x power

Question

for my homework it Says I need to sketch the graph of each function, so I need to graph this but I don't know how because I'm just learning this and I don't really understand especially with fractions and I'll appreciate if someone answers this

Answer 1

The first thing you'll want to do is think about the parent function of your equation. In this case, that's:

 

y = 4^x

 

It's the equation before any transformations (shifts or stretches or reflections) were applied to it.

 

This is an exponential equation, so as x becomes smaller and smaller (as we go left on the x-axis), the function gets closer and closer to 0. As x becomes larger and larger (as we go right on the x-axis), the function grows exponentially towards infinity. It's a good idea to know the general shape of the parent function, because your function will have the same general shape.

 

Yohan (below) actually graphed the parent function for you, in blue. So go check that out!

 

 

 

Now, you'll want to pick some anchor points from your parent function. Those are easy-to-work-with points that give you an idea of the parent function's shape. Plug in x = 0 and x = 1 to find two good anchor points for this one:

 

y = 4⁰

y = 1

 

That point is (0, 1).

 

y = 4¹

y = 4

 

That point is (1, 4).

 

In fact, for y = a^x, you can always use the points (0, 1) and (1, a) as your anchor points.

 

This means that the points (0, 1) and (1, 4) are on our parent function.

 

 

 

Next, you'll want to figure out what transformations were applied to the parent function to create your function. All that's new is that 1/2. Since it's being multiplied and isn't attached to x, that represents a vertical (not attached to x) stretch (multiplied) of 1/2. In other words, the function has been squashed vertically so that every point is only 1/2 as far from the x-axis as it was in the parent function.

 

Perform that transformation on your two points. A vertical stretch of 1/2 means multiply each point's y-coordinate by 1/2:

 

(0, 1) --> (0, 1/2)

(1, 4) --> (1, 2)

 

 

 

Finally, plot your new anchor points (0, 1/2) and (1, 2). Those give you a general idea of your new function. Remembering the general shape of the parent function (which is also the general shape of yours), and the fact that your new function has been squashed vertically, fill in the rest of the curve.

 

 

 

If you're still not quite sure what your function looks like, transform a few more anchor points:

 

y = 4²

y = 8

 

(2, 8) --> (2, 4)

 

y = 4^-1

y = 1/4

 

(-1, 1/4) --> (-1, 1/8)

 

y = 4³

y = 64

 

(3, 64) --> (3, 32)

 

You can plot (2, 4) and (-1, 1/8) on your graph for a better sense of its shape. (3, 32) probably won't fit on your graph, but that does tell you that the function is growing so fast that it has left your graph paper before you've gotten to 3 on the x-axis.

 

 

 

So, in general, your steps are:

 

1. Figure out the parent function and the parent function's general shape

2. Figure out two or three (or four) anchor points from the parent function

3. Figure out what transformations have been applied to your function

4. Apply those transformations to the parent function's anchor points

5. Plot the transformed anchor points and fill out your function's graph

Answer 2

Hi Emily, since none of us (tutors) actually graph for you, so here is the graph of 4 to the x power from the web:

 

I'm giving you this because it will give you a clue and basic shape of the graph:

 

 

http://www.mathipedia.com/aa12-function-transformations-exponential-and-logarithmic-equations.html

(Look the graphs with blue.  The one with red is transformation from blue by shifting.)

 

Your graph will look flatter because of 1/2 times the 4^x.  Blue is 4^x.  If the number in the front of 4^x is an integer or whole number that is greater than 1, the graph will go more fast and rapidly.  If the number is less than 1 (fraction), than graph will look more flatter and will go more slower.

 

I hope I answered or clarified some.  Look at the website that I gave you.

 

Yohan Chu   (LV  NV)