Quadratic Functions and Their Features

Definition and Key Vocabulary

Quadratic functions are a fundamental topic in College Algebra, characterized by their unique graph shapes and properties. Understanding their vocabulary is essential for graphing and solving related problems.

• Quadratic function: A function of the form with . The highest power of is 2.

• Parabola: The graph of a quadratic function. It appears as a smooth bowl (opens up) or an upside-down bowl (opens down).

• Vertex: The turning point of the parabola. It is the lowest point if the graph opens up, and the highest if it opens down.

• Axis of symmetry: The vertical line passing through the vertex, dividing the parabola into two mirror-image halves.

• Leading coefficient (a): The coefficient of in . Its sign determines the opening direction of the parabola.

• Vertex form: . The vertex is .

• General (standard) form: . The coefficients , , and define the parabola's shape and position.

• x-intercept: A point where the graph meets the x-axis. Found by solving .

• Zero or root: An input such that . Real zeros correspond to x-intercepts.

• y-intercept: The point . Substitute to find it; in general form, it is .

• Domain: All allowed -values. For unrestricted quadratics, the domain is .

• Range: All possible -values. Determined by the vertex height and the opening direction.

• Minimum or maximum: The smallest or largest output, which is the vertex's -value at .

Formulas and Graphing Steps

Vertex and Intercepts

To analyze and graph a quadratic function, follow these steps and formulas:

• Vertex from vertex form: gives vertex ; axis of symmetry .

• Vertex from general form: , . The vertex is .

• Opening direction: means the parabola opens up (minimum); means it opens down (maximum).

• Range: Opens up: ; opens down: .

• x-intercepts: Solve by factoring, square roots, or the quadratic formula.

• Quadratic formula: .

• Number of real zeros: Discriminant : (two zeros), (one repeated zero), (no real zeros).

• y-intercept: Compute ; point is .

Graphing Routine

1. Identify the form and coefficient to determine opening direction.

2. Find the vertex and draw the axis of symmetry.

3. Solve for real x-intercepts.

4. Find the y-intercept by substituting .

5. Plot the vertex and intercepts; add a reflected point if needed, then draw a smooth curve.

6. State domain, range, and minimum or maximum as required.

Common Mistakes

• Sign inside parentheses: means .

• Missing coefficient: means .

• Vertex formula: Divide by the entire .

• Square roots: Include both and when solving positive number.

• Answer format: Zero is an x-value; intercept is a point; axis is a line ; minimum/maximum value is at .

• Infinity always uses a parenthesis.

Shape reminder: Compared with at the same vertex, is narrower, is wider.

Worked Example: Vertex Form

Example:

This example demonstrates how to analyze and graph a quadratic in vertex form.

• Step 1: Identify , , . The graph opens down. Vertex: . Axis: .

• Step 2: Find x-intercepts by solving :

  • or

  • x-intercepts: and

• Step 3: Find y-intercept: . Point: . Reflection: .

• Step 4: Domain: . Range: . Maximum value: $4x = 1$.

Identification check: Point to the vertex, both x-intercepts, the y-intercept, and the axis of symmetry. Cover the labels and identify them again.

Worked Example: General Form

Example:

This example demonstrates how to analyze and graph a quadratic in general form.

• Step 1: Find the vertex:

  • , ,

  • Vertex: ; Axis:

• Step 2: Find intercepts:

  • x-intercepts: Use quadratic formula:

  • Intercepts: and (approx. and )

  • y-intercept: , so . Reflection: .

• Step 3: Domain: . Range: . Maximum value: $5x = 2$.

Self-check: The two real zeros average to $2$, the x-coordinate of the vertex. Both intercepts are equidistant from the axis.

Minimum, Maximum, and Applications

Finding Minimum or Maximum Values

The vertex provides the minimum or maximum value of a quadratic function, which is often used in optimization problems.

• Example 1:

  • (opens up, minimum)

  • Minimum value: $984x = 2(2, 984)$

  • Domain: ; Range:

• Example 2: Maximum area with 120 feet of fencing

  • Let = width, = length; Area

  • Perimeter:

  • Area function: (, maximum)

  • Best width: feet

  • Other dimension: feet

  • Maximum area: square feet

Recognizing optimization questions: Look for phrases like largest area, smallest cost, highest value, maximum, or minimum. Define the quantity, write a function in one variable, find the vertex, and interpret both coordinates with units.

Context: Word problems may restrict the domain (e.g., for rectangle dimensions).

Practice and Readiness Check

Vocabulary and Identification

• Axis of symmetry: The vertical mirror line of a parabola.

• Zero: An input that makes equal zero.

• Range: The set of possible output values.

• Vertex: The turning point of the graph.

• y-intercept: The point found by setting .

• Domain: The set of permitted input values.

Graphing and Problem Solving

• Identify , , , vertex, axis, minimum/maximum for given functions.

• Find intercepts, domain, range, and plot points for various quadratic functions.

• Apply the quadratic formula to find real zeros and corresponding intercepts.

• Explain why some quadratics have no real x-intercepts (e.g., ).

Applications and Error Checking

• Model area problems with quadratic functions and interpret answers with units.

• Correct common errors in vertex identification and maximum/minimum values.

• State axis, domain, range, and minimum/maximum for given vertex and opening direction.

Summary Table: Quadratic Function Properties

Readiness Checklist

• Identify vocabulary in equations and graphs

• Find the full vertex from either equation form

• Find intercepts and sketch a parabola

• State domain, range, and minimum/maximum correctly

• Model an area problem and explain the answer with units

Next step: Review missed skills, study worked examples, and solve similar problems independently.