Quadratic Functions and Vertex Form

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is a useful way to express the equation of a parabola, especially when the vertex and another point are known. The general vertex form is:

• Formula: where is the vertex of the parabola.

• Application: If you know the vertex and one additional point on the graph, you can solve for and write the quadratic function.

Example: Determine the quadratic function whose graph has a vertex of and passes through the point .

Maximum and Minimum Values of Quadratic Functions

The direction in which a parabola opens depends on the sign of the coefficient in the vertex form.

• If , the vertex is the minimum value of the function, and the parabola opens upward.

• If , the vertex is the maximum value of the function, and the parabola opens downward.

• The maximum or minimum value of is .

Optimization with Quadratic Functions

Optimization Problems

Quadratic functions are often used to model real-world situations where you need to find maximum or minimum values, such as maximizing area, profit, or height. The vertex of the parabola gives the location of the maximum or minimum value.

• Steps for Optimization:

  1. Express the situation as a quadratic function.

  2. Identify the vertex to find the maximum or minimum value.

  3. Interpret the vertex in the context of the problem.

• Key Point: The -coordinate of the vertex gives the location of the max/min, and the -coordinate gives the value.

Example: Projectile Motion

A projectile is fired from a cliff 500 feet above water at an inclination of with an initial velocity of 100 ft/sec. The height of the projectile above the water is given by:

• Formula: where is the time in seconds after the projectile leaves the base of the cliff.

• (a) Find the maximum height of the projectile. Solution: The maximum height occurs at the vertex. The -coordinate of the vertex is for . seconds Maximum height: feet

• (b) How far from the base of the cliff will the projectile hit the water? Solution: Set and solve for : Solve for (using quadratic formula): seconds Horizontal distance: feet

• (c) When the height of the projectile is 1000 feet, how far is it from the base of the cliff? Solution: Set and solve for : Solve for (using quadratic formula): seconds Horizontal distance: feet

Applications: Maximizing Area

Example: Maximizing the Area of a Rectangle

A farmer has 2000 yards of fence to enclose a rectangular field. What are the dimensions that enclose the most area?

• (a) Express the area of the rectangle as a function of the width . Solution: Let be the length. The perimeter is , so . Area:

• (b) For what value of is the area largest? Solution: The maximum area occurs at the vertex: yards

• (c) What is the maximum area? Solution: square yards

Applications: Revenue Maximization

Revenue Function in Economics

In economics, Revenue is defined as the amount of money received from the sale of an item. It is the product of the unit price and the number of units sold.

• Formula: where is revenue, is price, and is quantity sold.

Example: Maximizing Revenue

The price (in dollars) and the quantity sold of a certain product obey the demand equation:

• Demand Equation:

• (a) Find a model that expresses the revenue as a function of . What is the domain of ? Solution: Domain: (quantity sold must be positive)

• (b) What is the revenue if 100 units are sold? Solution: dollars

• (c) What is the quantity that maximizes the revenue? What is the maximum revenue? Solution: To maximize , set the derivative and solve for . Set ; solve for . Maximum revenue occurs at units. Maximum revenue: dollars

• (d) What price should the company charge to maximize the revenue? Solution: Substitute into the demand equation: dollars per unit

Summary Table: Quadratic Applications