Question
Tom started an entertainment company. The net value of the company (in thousands of dollars)
\[t\] months after its creation is modeled by
\[v(t)=4t^2-24t-28\]
Tom wants to know when his company will be at its lowest net value.
1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation.
\[v(t)=\]
2) How many months after its creation does the company reach its lowest net value?
months
\[t\] months after its creation is modeled by
\[v(t)=4t^2-24t-28\]
Tom wants to know when his company will be at its lowest net value.
1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation.
\[v(t)=\]
2) How many months after its creation does the company reach its lowest net value?
months
Answers
GPT-4o mini
To find when Tom's entertainment company reaches its lowest net value using the function \( v(t) = 4t^2 - 24t - 28 \), we will rewrite the equation in vertex form and find the vertex.
1) **Rewriting the function in vertex form:**
The vertex form of a quadratic function \( v(t) = a(t - h)^2 + k \) allows us to easily find the vertex \((h,k)\). We can start by completing the square for the quadratic \(v(t)\):
1. Factor out the coefficient of \(t^2\) (which is 4) from the first two terms:
\[
v(t) = 4(t^2 - 6t) - 28
\]
2. Next, to complete the square inside the parentheses, we take half of the coefficient of \(t\) (which is -6), square it, and add and subtract that square:
\[
\left(-\frac{6}{2}\right)^2 = 9
\]
3. We add and subtract 9 inside the parentheses:
\[
v(t) = 4(t^2 - 6t + 9 - 9) - 28
\]
This simplifies to:
\[
v(t) = 4((t - 3)^2 - 9) - 28
\]
4. Expanding this gives:
\[
v(t) = 4(t - 3)^2 - 36 - 28
\]
\[
v(t) = 4(t - 3)^2 - 64
\]
Now, we have the function in vertex form:
\[
\boxed{v(t) = 4(t - 3)^2 - 64}
\]
2) **Finding the time when the company reaches its lowest net value:**
The minimum value of a parabola \(a(t - h)^2 + k\) occurs at \(t = h\). From our vertex form \(v(t) = 4(t - 3)^2 - 64\), we see that:
\[
h = 3
\]
Thus, the company reaches its lowest net value:
\[
\boxed{3} \text{ months}
\] after its creation.
1) **Rewriting the function in vertex form:**
The vertex form of a quadratic function \( v(t) = a(t - h)^2 + k \) allows us to easily find the vertex \((h,k)\). We can start by completing the square for the quadratic \(v(t)\):
1. Factor out the coefficient of \(t^2\) (which is 4) from the first two terms:
\[
v(t) = 4(t^2 - 6t) - 28
\]
2. Next, to complete the square inside the parentheses, we take half of the coefficient of \(t\) (which is -6), square it, and add and subtract that square:
\[
\left(-\frac{6}{2}\right)^2 = 9
\]
3. We add and subtract 9 inside the parentheses:
\[
v(t) = 4(t^2 - 6t + 9 - 9) - 28
\]
This simplifies to:
\[
v(t) = 4((t - 3)^2 - 9) - 28
\]
4. Expanding this gives:
\[
v(t) = 4(t - 3)^2 - 36 - 28
\]
\[
v(t) = 4(t - 3)^2 - 64
\]
Now, we have the function in vertex form:
\[
\boxed{v(t) = 4(t - 3)^2 - 64}
\]
2) **Finding the time when the company reaches its lowest net value:**
The minimum value of a parabola \(a(t - h)^2 + k\) occurs at \(t = h\). From our vertex form \(v(t) = 4(t - 3)^2 - 64\), we see that:
\[
h = 3
\]
Thus, the company reaches its lowest net value:
\[
\boxed{3} \text{ months}
\] after its creation.