To determine the general form of the equation that best fits the given data points, we will analyze the data and derive the proper model.
Given data points:
- (-3, [tex]\( \frac{1}{8} \)[/tex])
- (-2, [tex]\( \frac{1}{4} \)[/tex])
- (-1, [tex]\( \frac{1}{2} \)[/tex])
- (0, 1)
We are considering the following general forms:
1. [tex]\( y = mx + b \)[/tex] - Linear equation
2. [tex]\( y = a b^x \)[/tex] - Exponential equation
3. [tex]\( y = ax^2 + bx + c \)[/tex] - Quadratic equation
First, let's observe the pattern in the given [tex]\( y \)[/tex]-values as [tex]\( x \)[/tex]-values change.
Pattern Recognition:
Transforming the [tex]\( y \)[/tex]-values into a more recognizable form by rewriting them in terms of powers of 2:
- [tex]\( \frac{1}{8} = 2^{-3} \)[/tex]
- [tex]\( \frac{1}{4} = 2^{-2} \)[/tex]
- [tex]\( \frac{1}{2} = 2^{-1} \)[/tex]
- [tex]\( 1 = 2^{0} \)[/tex]
Here, it is evident that the [tex]\( y \)[/tex]-values can be represented as [tex]\( y = 2^x \)[/tex]. This suggests an exponential relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex].
Thus, the general form of the equation that represents the data is:
[tex]\[ y = a b^x \][/tex]
To find the constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we consider the table and note that the form [tex]\( y = ab^x \)[/tex] matches the data perfectly if [tex]\( a = 1 \)[/tex] and [tex]\( b = 2 \)[/tex]:
- For [tex]\(x = 0, y = 1\)[/tex], which confirms [tex]\( a = 1 \)[/tex].
- For other values of [tex]\(x\)[/tex], we observe the exponential growth factor [tex]\( b = 2 \)[/tex].
So, the equation that represents the data is:
[tex]\[ y = 1 \cdot 2^x \][/tex]
Simplifying, we get:
[tex]\[ y = 2^x \][/tex]
Therefore, Kareem can use the exponential form [tex]\( y = ab^x \)[/tex] to represent the data, where [tex]\( a = 1 \)[/tex] and [tex]\( b = 2 \)[/tex]. Thus, the specific equation is [tex]\( y = 2^x \)[/tex].
