Math  /  Algebra

QuestionThe figure shows a portion of the graph of a function ff. Which of the following conclusions is possible for f? (A) ff is a quadratic function because the output values are proportional over equal-length input-value intervals. (B) ff is a quadratic function because the average rates of change over consecutive equallength input-value intervals can be given by a linear function. (C) ff is an exponential function because the output values are proportional over equallength input-value intervals. (D) ff is an exponential funotion because the average rates of change over consecutive equal-ength inpusvalue intervals can be given by a linear function.

Studdy Solution

STEP 1

What is this asking? Which type of function, quadratic or exponential, could the graph represent, and why? Watch out! Don't just assume it's exponential because it looks like it; we need to check the properties!

STEP 2

1. Analyze Quadratic Functions
2. Analyze Exponential Functions
3. Analyze the Graph
4. Determine the Function Type

STEP 3

Quadratic functions have a constant second difference.
This means that the difference of the differences between consecutive outputs over equal intervals of input should be the same.
Let's see what that means!

STEP 4

Imagine a simple quadratic function, like f(x)=x2f(x) = x^2.
Let's pick some equally spaced xx values, say 0,1,2,30, 1, 2, 3.

STEP 5

Now, let's calculate the corresponding yy values: f(0)=0f(0) = 0, f(1)=1f(1) = 1, f(2)=4f(2) = 4, f(3)=9f(3) = 9.

STEP 6

The differences between consecutive yy values are 10=11 - 0 = 1, 41=34 - 1 = 3, 94=59 - 4 = 5.
Notice how these differences are not constant.

STEP 7

Let's find the differences of *these* differences: 31=23 - 1 = 2, 53=25 - 3 = 2.
See? *These* differences are constant!
That's what we mean by a constant second difference.

STEP 8

Exponential functions have a constant ratio between consecutive outputs over equal intervals of input.
This means if we divide consecutive yy values, we should always get the same number.

STEP 9

Think about f(x)=2xf(x) = 2^x.
Let's use the same xx values: 0,1,2,30, 1, 2, 3.

STEP 10

The yy values are f(0)=1f(0) = 1, f(1)=2f(1) = 2, f(2)=4f(2) = 4, f(3)=8f(3) = 8.

STEP 11

Now, divide consecutive yy values: 21=2\frac{2}{1} = 2, 42=2\frac{4}{2} = 2, 84=2\frac{8}{4} = 2.
See? The ratio is always **2**!

STEP 12

The graph curves upwards, so it *could* be either quadratic or exponential.
We need to look closer!

STEP 13

Let's pick some points on the graph with equally spaced xx values.
It looks like the graph goes through points that are *roughly* at (0,1)(0, 1), (1,2)(1, 2), (2,4)(2, 4), and (3,8)(3, 8).

STEP 14

The yy values are roughly 1,2,4,81, 2, 4, 8.
Let's check the ratios: 21=2\frac{2}{1} = 2, 42=2\frac{4}{2} = 2, 84=2\frac{8}{4} = 2.
The ratio is constant!

STEP 15

Now let's check the differences: 21=12 - 1 = 1, 42=24 - 2 = 2, 84=48 - 4 = 4.
The differences of the differences are 21=12 - 1 = 1 and 42=24 - 2 = 2.
Not constant!

STEP 16

Since the ratio of consecutive outputs is constant, and the second difference isn't, the graph likely represents an exponential function.

STEP 17

Option (C) says the function is exponential because the output values are proportional, which matches our observation of a constant ratio.

STEP 18

The answer is (C).

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