Question
The price of a certain stock, in dollars, can be modeled by the function f(x)=0.75x^2-6x+20, where x equals the number of months that the stock is on the exchange. Over the same 6-month period, the price of another stock, in dollars, increases according to the function g(x)=2.75+1.50. graph the functions in the same coordinate plane to determine when f(x)=g(x). How many times during the 6-month period will both stocks be at the same price?
1)three times
2)once
3)never
4)twice
1)three times
2)once
3)never
4)twice
Answers
GPT-4o mini
First, let's clarify the functions given in the problem.
1. The price of the first stock is modeled by the function:
\[
f(x) = 0.75x^2 - 6x + 20
\]
2. The price of the second stock is given as:
\[
g(x) = 2.75 + 1.50
\]
This simplifies to:
\[
g(x) = 4.25
\]
We need to find out when \( f(x) = g(x) \), which translates to solving the equation:
\[
0.75x^2 - 6x + 20 = 4.25
\]
Subtract \( 4.25 \) from both sides:
\[
0.75x^2 - 6x + 20 - 4.25 = 0
\]
\[
0.75x^2 - 6x + 15.75 = 0
\]
Next, we can multiply the entire equation by 4 to eliminate the decimal:
\[
3x^2 - 24x + 63 = 0
\]
Now let's solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 3 \), \( b = -24 \), and \( c = 63 \).
First, calculate the discriminant \( b^2 - 4ac \):
\[
b^2 = (-24)^2 = 576
\]
\[
4ac = 4 \cdot 3 \cdot 63 = 756
\]
\[
b^2 - 4ac = 576 - 756 = -180
\]
Since the discriminant is negative (\(-180\)), this indicates that the quadratic equation has no real solutions.
As a result, the graphs of \( f(x) \) and \( g(x) \) do not intersect.
Therefore, the answer to how many times during the 6-month period will both stocks be at the same price is:
3) never.
1. The price of the first stock is modeled by the function:
\[
f(x) = 0.75x^2 - 6x + 20
\]
2. The price of the second stock is given as:
\[
g(x) = 2.75 + 1.50
\]
This simplifies to:
\[
g(x) = 4.25
\]
We need to find out when \( f(x) = g(x) \), which translates to solving the equation:
\[
0.75x^2 - 6x + 20 = 4.25
\]
Subtract \( 4.25 \) from both sides:
\[
0.75x^2 - 6x + 20 - 4.25 = 0
\]
\[
0.75x^2 - 6x + 15.75 = 0
\]
Next, we can multiply the entire equation by 4 to eliminate the decimal:
\[
3x^2 - 24x + 63 = 0
\]
Now let's solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 3 \), \( b = -24 \), and \( c = 63 \).
First, calculate the discriminant \( b^2 - 4ac \):
\[
b^2 = (-24)^2 = 576
\]
\[
4ac = 4 \cdot 3 \cdot 63 = 756
\]
\[
b^2 - 4ac = 576 - 756 = -180
\]
Since the discriminant is negative (\(-180\)), this indicates that the quadratic equation has no real solutions.
As a result, the graphs of \( f(x) \) and \( g(x) \) do not intersect.
Therefore, the answer to how many times during the 6-month period will both stocks be at the same price is:
3) never.