Question
Y=t^2+3t ; Find g(-6)
Answer should be 18
Answer should be 18
Answers
Answer
Why is this a function but not linear?
Answered by
GPT-4o mini
It seems there might be a small misunderstanding regarding the function provided and the required value. The equation given is \( Y = t^2 + 3t \), which is a quadratic function of \( t \).
To find \( g(-6) \), we assume that \( g(t) \) refers to the same function \( Y(t) = t^2 + 3t \). To find \( g(-6) \), we simply substitute \( t = -6 \) into the equation:
\[
g(-6) = (-6)^2 + 3(-6)
\]
Calculating step-by-step:
1. \( (-6)^2 = 36 \)
2. \( 3 \times (-6) = -18 \)
Now, we can combine these results:
\[
g(-6) = 36 - 18 = 18
\]
Thus, the answer is indeed \( g(-6) = 18 \).
To find \( g(-6) \), we assume that \( g(t) \) refers to the same function \( Y(t) = t^2 + 3t \). To find \( g(-6) \), we simply substitute \( t = -6 \) into the equation:
\[
g(-6) = (-6)^2 + 3(-6)
\]
Calculating step-by-step:
1. \( (-6)^2 = 36 \)
2. \( 3 \times (-6) = -18 \)
Now, we can combine these results:
\[
g(-6) = 36 - 18 = 18
\]
Thus, the answer is indeed \( g(-6) = 18 \).
Answered by
GPT-4o mini
The function you provided, \( Y = t^2 + 3t \), is a quadratic function, which is a specific type of polynomial function. Here's why it is classified as a quadratic function and not a linear function:
### Characteristics of Linear Functions:
1. **Form**: A linear function has the form \( Y = mt + b \), where \( m \) and \( b \) are constants.
2. **Degree**: The highest power of the variable \( t \) is 1 (i.e., it is of degree 1).
3. **Graph**: The graph of a linear function is a straight line.
4. **Rate of Change**: The rate of change is constant; for every unit increase in \( t \), \( Y \) changes by a constant amount.
### Characteristics of Quadratic Functions:
1. **Form**: A quadratic function has the form \( Y = at^2 + bt + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).
2. **Degree**: The highest power of the variable \( t \) is 2 (i.e., it is of degree 2).
3. **Graph**: The graph of a quadratic function is a parabola. It can open upwards or downwards depending on the sign of the coefficient \( a \).
4. **Rate of Change**: The rate of change is not constant; it changes as \( t \) changes. This is because the term \( t^2 \) causes the slope of the function to vary depending on the value of \( t \).
### In Your Case:
In the function \( Y = t^2 + 3t \):
- The highest power of \( t \) is 2 (from the \( t^2 \) term), which means it is a quadratic function.
- Consequently, this function will produce a parabolic graph rather than a straight line, confirming that it is not linear.
In summary, the presence of the \( t^2 \) term distinguishes the function as quadratic, while a lack of such a term would indicate a linear function.
### Characteristics of Linear Functions:
1. **Form**: A linear function has the form \( Y = mt + b \), where \( m \) and \( b \) are constants.
2. **Degree**: The highest power of the variable \( t \) is 1 (i.e., it is of degree 1).
3. **Graph**: The graph of a linear function is a straight line.
4. **Rate of Change**: The rate of change is constant; for every unit increase in \( t \), \( Y \) changes by a constant amount.
### Characteristics of Quadratic Functions:
1. **Form**: A quadratic function has the form \( Y = at^2 + bt + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).
2. **Degree**: The highest power of the variable \( t \) is 2 (i.e., it is of degree 2).
3. **Graph**: The graph of a quadratic function is a parabola. It can open upwards or downwards depending on the sign of the coefficient \( a \).
4. **Rate of Change**: The rate of change is not constant; it changes as \( t \) changes. This is because the term \( t^2 \) causes the slope of the function to vary depending on the value of \( t \).
### In Your Case:
In the function \( Y = t^2 + 3t \):
- The highest power of \( t \) is 2 (from the \( t^2 \) term), which means it is a quadratic function.
- Consequently, this function will produce a parabolic graph rather than a straight line, confirming that it is not linear.
In summary, the presence of the \( t^2 \) term distinguishes the function as quadratic, while a lack of such a term would indicate a linear function.