Question

1. write the quadratic equation that models the restaurants profit in vertex form. ( y = a(x - h)^2 + k )( y = -\frac{1}{2}(x - 30)^2 + 450 )4. rewrite the equation in standard form.

Explanation:

Step1: Expand the squared term

We start with the vertex form of the quadratic equation \( y = -\frac{1}{2}(x - 30)^2 + 450 \). First, we expand \( (x - 30)^2 \) using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \), where \( a = x \) and \( b = 30 \). So, \( (x - 30)^2 = x^2 - 60x + 900 \).
Now our equation becomes \( y = -\frac{1}{2}(x^2 - 60x + 900) + 450 \).

Step2: Distribute the coefficient

Next, we distribute the \( -\frac{1}{2} \) across the terms inside the parentheses.
\( -\frac{1}{2} \times x^2 = -\frac{1}{2}x^2 \),
\( -\frac{1}{2} \times (-60x) = 30x \),
\( -\frac{1}{2} \times 900 = -450 \).
So after distributing, the equation is \( y = -\frac{1}{2}x^2 + 30x - 450 + 450 \).

Step3: Simplify the constant terms

Now we simplify the constant terms \( -450 + 450 \), which equals 0. So the equation simplifies to \( y = -\frac{1}{2}x^2 + 30x \).

Answer:

The standard form of the quadratic equation is \( y = -\frac{1}{2}x^2 + 30x \)