Lesson 4: Modeling with Quadratic Functions

Property

Use a systematic problem-solving strategy for quadratic modeling applications:
Step 1. Read the problem to understand the context and relationships.
Step 2. Identify what you are looking for (maximum, minimum, dimensions, etc.).
Step 3. Define variables for unknown quantities.
Step 4. Translate the problem into a quadratic equation or function.
Step 5. Solve the equation or analyze the function as needed.
Step 6. Check if the answer makes sense in the problem context.
Step 7. Answer the question with a complete sentence including appropriate units.

Examples

Property

For a rectangle with length, $L$, and width, $W$, the area, $A$, is given by the formula:

In application problems, one dimension is often expressed in terms of the other (e.g., $L = 3W-1$).
Substituting this into the area formula creates a quadratic equation.
If the equation cannot be factored, use the Quadratic Formula to find the solution and round as needed.

Examples

• A rectangular garden has an area of 100 square feet. Its length is 15 feet more than its width. Find the dimensions. Let the width be $W$. The equation is $100 = W(W+15)$, or $W^2 + 15W - 100 = 0$. Factoring gives $(W+20)(W-5)=0$. The width is 5 feet and the length is 20 feet.

• The length of a rectangular patio is 1 foot less than twice its width. The area is 105 square feet. Find the length and width. Let the width be $W$. The equation is $105 = W(2W-1)$, or $2W^2 - W - 105 = 0$. Factoring gives $(2W+15)(W-7)=0$. The width is 7 feet and the length is 13 feet.

Property

The height in feet, $h$, of an object shot upwards into the air with initial velocity, $v_0$, from an initial height $h_0$, after $t$ seconds is given by the formula:

Examples