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The points $(0,1)$ and $(1,4)$ are contained in the graph of the equation with only two variables, $x$ and $y$. Select all of the true statements.


  • There is more than one equation in the form of $y=a(b)^x$ that contains these points.
  • There is exactly one equation in the form of $y=a(b)^x$ that contains these points.
  • There is exactly one equation in the form of $y=m x+b$ that contains these points.
  • There are no equations in the form of $y = mx + b$ that contain these points.
  • There are two equations in the form of $y=m x+b$ that contain these points.

Verified Answer :

  • Substituting the points \((0, 1)\) and \((1, 4)\) into the form \(y = a(b)^x\) yields the unique equation \(y = 4^x\).
  • Substituting the same points into the form \(y = mx + b\) yields the unique linear equation \(y = 3x + 1\).
  • Consequently, exactly one equation exists for each form.
  • The true statements are:
  1. There is exactly one equation in the form of \(y = a(b)^x\) that contains these points.
  2. There is exactly one equation in the form of \(y = mx + b\) that contains these points.

Explanation

  1. Analyzing the Equations for Given Points

We are given two points, \((0, 1)\) and \((1, 4)\), and we need to determine which statements are true regarding equations of the forms \(y = a(b)^x\) and \(y = mx + c\) that pass through these points.

First, let's analyze the exponential form \(y = a(b)^x\).
Substituting the point \((0, 1)\):
\[1 = a(b)^0 = a(1) = a\]
So, \(a = 1\).
Substituting the point \((1, 4)\):
\[4 = a(b)^1 = 1(b) = b\]
So, \(b = 4\).
The unique equation is \(y = 1(4)^x = 4^x\). Therefore, there is exactly one equation of the form \(y = a(b)^x\) passing through these points.

Next, let's analyze the linear form \(y = mx + b\) (we will use \(c\) as the intercept to distinguish it from the \(b\) in the exponential form: \(y = mx + c\)).
Substituting the point \((0, 1)\):
\[1 = m(0) + c \implies c = 1\]
Substituting the point \((1, 4)\):
\[4 = m(1) + 1 \implies m + 1 = 4 \implies m = 3\]
The unique equation is \(y = 3x + 1\). Therefore, there is exactly one equation of the form \(y = mx + b\) passing through these points.

  1. Evaluating the Statements

Based on our analysis:

  • The statement "There is exactly one equation in the form of \(y = a(b)^x\) that contains these points" is True.
  • The statement "There is exactly one equation in the form of \(y = mx + b\) that contains these points" is True.
  • The other statements, which suggest there is more than one or no such equations, are False.

Examples

This problem illustrates how two distinct points uniquely define both a linear relationship and an exponential relationship. In real-world applications, this is fundamental to data fitting. For instance, if you track the growth of a bank balance over two months, you can use these mathematical models to predict future values. A linear model might represent simple interest, while an exponential model represents compound interest, helping you understand how different growth rates affect your savings over time.

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