Question
Y=-3times2^x ; Find f(1)
Answer should be -6
Answer should be -6
Answers
Answered by
GPT-4o mini
To find \( f(1) \) for the function \( Y = -3 \times 2^x \), you need to substitute \( x \) with 1.
\[
f(1) = -3 \times 2^1
\]
Calculating this gives:
\[
f(1) = -3 \times 2 = -6
\]
So, your answer is indeed \( -6 \).
\[
f(1) = -3 \times 2^1
\]
Calculating this gives:
\[
f(1) = -3 \times 2 = -6
\]
So, your answer is indeed \( -6 \).
Answer
Why is this equation not linear?
Answered by
GPT-4o mini
The equation \( y = -3 \times 2^x \) is not linear for several reasons:
1. **Exponential Nature**: The term \( 2^x \) indicates that the function grows (or decays, since it's multiplied by \(-3\)) exponentially rather than linearly. In a linear function, the variable \( x \) is raised to the first power (i.e., \( y = mx + b \)), while in this case, \( x \) is found in the exponent.
2. **Graph Shape**: The graph of an exponential function will exhibit a curved shape, with rapid increases or decreases, rather than the straight line that characterizes linear functions. Specifically, as \( x \) increases, \( 2^x \) grows exponentially, which means that \( y \) will also change at a non-constant rate.
3. **Rate of Change**: In a linear function, the rate of change (slope) is constant. In this exponential function, the rate of change is not constant—it increases as \( x \) increases. The slope of the tangent line at any point on the curve changes depending on the value of \( x \).
These characteristics—exponential growth, curved graph, and variable rate of change—are key distinctions that demonstrate why the equation \( y = -3 \times 2^x \) is classified as an exponential function rather than a linear function.
1. **Exponential Nature**: The term \( 2^x \) indicates that the function grows (or decays, since it's multiplied by \(-3\)) exponentially rather than linearly. In a linear function, the variable \( x \) is raised to the first power (i.e., \( y = mx + b \)), while in this case, \( x \) is found in the exponent.
2. **Graph Shape**: The graph of an exponential function will exhibit a curved shape, with rapid increases or decreases, rather than the straight line that characterizes linear functions. Specifically, as \( x \) increases, \( 2^x \) grows exponentially, which means that \( y \) will also change at a non-constant rate.
3. **Rate of Change**: In a linear function, the rate of change (slope) is constant. In this exponential function, the rate of change is not constant—it increases as \( x \) increases. The slope of the tangent line at any point on the curve changes depending on the value of \( x \).
These characteristics—exponential growth, curved graph, and variable rate of change—are key distinctions that demonstrate why the equation \( y = -3 \times 2^x \) is classified as an exponential function rather than a linear function.