Question

1. give the coordinates of the vertex of m(x)=-9(x + 4)^2+5. what is the maximum or minimum value? 4. same as one of the problems in wiley for the function f(x)=a(x - h)^2 + k graphed below, state whether the constants h and k are positive, negative or zero. 5. think backwards from the above problems. the graphs of a quadratic function are given below. find a possible formula for each function. assume a is ±1. 6. same as one of the problems in wiley the profit (in thousands of dollars) a company makes from selling a certain item depends on the price of the item. the three different forms for the profit at a price of p dollars are: standard form: -2p^2+24p - 54 factored form: -2(p - 3)(p - 9) vertex form: -2(p - 6)^2+18 a) sketch a rough graph of the function, labeling your axes: b) which form is most useful for finding the prices that give a profit of zero dollars? (these are called the break - even prices.) use it to find these prices. form: ____________ prices (with units): __________ c) which form is most useful for finding the profit when the price is zero? use it to find that profit. form: __________ profit when price is zero (with units): __________ what does it mean for profit to be negative? does that make sense? d) the company would like to maximize profits. which form is most useful for finding the price that gives the maximum profit? use it to find the optimal price and the maximum profit. form: __________ optimal price (with units): __________ max profit (with units): ____________

Explanation:

Problem 1:

Step1: Identify vertex form

The function \(m(x)=-9(x + 4)^2+5\) is in \(y=a(x - h)^2+k\) form. Here \(h=-4\) and \(k = 5\).

Step2: Determine maximum/minimum

Since \(a=-9<0\), the parabola opens downwards, so the vertex is a maximum.

Problem 6:

a) Sketching:

Step1: Find vertex

Use the formula \(p=-\frac{b}{2a}\) for \(y=-2p^2+24p - 54\) to find \(p = 6\), then find \(y\) - value of vertex.

Step2: Find intercepts

Find \(y\) - intercept by setting \(p = 0\) in standard form and \(x\) - intercepts using factored form.

b) Break - even:

Step1: Choose form

The factored form allows us to set each factor equal to zero.

Step2: Solve for \(p\)

\(p-3 = 0\) or \(p - 9=0\) gives break - even prices.

c) Profit at \(p = 0\):

Step1: Choose form

Standard form is used to substitute \(p = 0\) directly.

Step2: Calculate profit

\(y=-2\times0^2+24\times0-54=-54\).

d) Maximum profit:

Step1: Choose form

Vertex form gives the maximum point directly for a quadratic with \(a<0\).

Step2: Identify optimal price and profit

The vertex \((6,18)\) gives optimal price \(p = 6\) and max profit \(y = 18\).

Answer:

Problem 1:

1. Coordinates of the vertex of \(m(x)=-9(x + 4)^2+5\):

• The vertex - form of a quadratic function is \(y=a(x - h)^2+k\), where the vertex is \((h,k)\). For the function \(m(x)=-9(x + 4)^2+5\), we can rewrite it as \(m(x)=-9(x-(-4))^2 + 5\).
• So the vertex is \((-4,5)\).
• Since \(a=-9<0\), the parabola opens downwards and the vertex represents the maximum value. The maximum value is \(y = 5\).

Problem 6:

a) Sketch of \(y=-2p^2 + 24p-54\):

• First, find the vertex. For a quadratic function in standard - form \(y = ax^2+bx + c\), the \(x\) - coordinate of the vertex is \(p=-\frac{b}{2a}\). Here, \(a=-2\), \(b = 24\), so \(p=-\frac{24}{2\times(-2)}=6\).
• Substitute \(p = 6\) into the function: \(y=-2\times6^2+24\times6 - 54=-72 + 144-54 = 18\). The vertex is \((6,18)\).
• The \(y\) - intercept is found by setting \(p = 0\): \(y=-54\).
• To find the \(x\) - intercepts, we can use the factored form \(-2(p - 3)(p - 9)=0\), so \(p = 3\) and \(p = 9\). Sketch a parabola opening downwards (because \(a=-2<0\)) passing through \((3,0)\), \((9,0)\), \((0,-54)\) and having vertex \((6,18)\).

b) Break - even prices:

• The factored form \(-2(p - 3)(p - 9)\) is most useful for finding the break - even prices.
• Set \(-2(p - 3)(p - 9)=0\). Using the zero - product property, \(p-3 = 0\) or \(p - 9=0\). So the break - even prices are \(p = 3\) dollars and \(p = 9\) dollars.
• Form: Factored form. Prices (with units): \(p = 3\) dollars, \(p = 9\) dollars.

c) Profit when price is zero:

• The standard form \(y=-2p^2+24p - 54\) is most useful. Substitute \(p = 0\) into the standard form: \(y=-2\times0^2+24\times0-54=-54\) (in thousands of dollars).
• Form: Standard form. Profit when price is zero (with units): \(-54\) thousand dollars. A negative profit means the company is making a loss. It makes sense because when the price is zero, the company is giving away the item and incurring costs.

d) Maximum profit:

• The vertex form \(-2(p - 6)^2+18\) is most useful.
• The vertex of the parabola in vertex form \(y=a(x - h)^2+k\) gives the maximum or minimum of the function. For \(y=-2(p - 6)^2+18\), since \(a=-2<0\), the vertex \((6,18)\) represents the maximum.
• The optimal price is \(p = 6\) dollars, and the maximum profit is \(18\) thousand dollars.
• Form: Vertex form. Optimal price (with units): \(p = 6\) dollars. Max profit (with units): \(18\) thousand dollars.