Math  /  Algebra

QuestionConsider the graph of the exponential function below.
Determine the rule of this function in the form y=cxy=c^{x}.

Studdy Solution

STEP 1

What is this asking? We need to find the equation of an exponential function, specifically in the form y=cxy = c^x, given a graph with some plotted points. Watch out! Don't mix up the x and y values!
Also, remember that a decreasing exponential function means our *c* value will be between 0 and 1!

STEP 2

1. Identify a Useful Point
2. Substitute and Solve
3. Verify with Another Point

STEP 3

Alright, let's pick a point from our graph!
We could choose any of them, but the point (1,34)(1, \frac{3}{4}) looks pretty friendly.
Using this point will make our calculations easier!

STEP 4

Remember, our goal is to find the equation in the form y=cxy = c^x.
We'll **substitute** our chosen x and y values into this equation.
So, we have 34=c1\frac{3}{4} = c^1.

STEP 5

Now, any number raised to the power of 1 is just itself!
So, c1=cc^1 = c.
That means 34=c\frac{3}{4} = c.
Wow, that was quick!
We found our **c value**: c=34c = \frac{3}{4}.

STEP 6

Let's double-check our *c* value using a different point to be extra sure!
Let's try (1,43)(-1, \frac{4}{3}).
Our equation is y=(34)xy = (\frac{3}{4})^x.

STEP 7

**Substitute** x=1x = -1 and y=43y = \frac{4}{3} into our equation: 43=(34)1\frac{4}{3} = (\frac{3}{4})^{-1}.

STEP 8

Remember, a negative exponent means we flip the fraction.
So, (34)1(\frac{3}{4})^{-1} becomes 43\frac{4}{3}.
This gives us 43=43\frac{4}{3} = \frac{4}{3}.
It matches!
Awesome!

STEP 9

Our equation is y=(34)xy = (\frac{3}{4})^x.
We found the *c* value and verified it.
We're done!

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