Quadratic Functions and Equations

Graphing Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written as y = ax^2 + bx + c. Their graphs are parabolas, which can open upwards or downwards depending on the sign of a.

• Axis of Symmetry: The vertical line that divides the parabola into two mirror images. It is given by .

• Vertex: The highest or lowest point of the parabola, located at where and is found by substituting this value into the function.

• Direction: If a > 0, the parabola opens upwards; if a < 0, it opens downwards.

• Intercepts: The y-intercept is at ; x-intercepts are found by solving .

Example: For , the axis of symmetry is , and the vertex is at .

Qualitative Graphing and Transformations

Transformations such as shifting, stretching, or reflecting affect the graph of a quadratic function. For example, is a vertical stretch and upward shift of .

• Vertical Stretch: Multiplying by a constant > 1 makes the parabola narrower.

• Vertical Shift: Adding a constant shifts the graph up or down.

Example: Graph by starting with , stretching vertically by 2, and shifting up by 4 units.

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula:

• Factoring: Express the quadratic as a product of binomials and set each factor to zero.

• Quadratic Formula:

• Completing the Square: Rearrange and solve by isolating the squared term.

Example: Solve by factoring: , so or .

Applications of Quadratic Functions

Quadratic functions model projectile motion, area problems, and optimization.

• Projectile Motion: Height as a function of time:

• Area Problems: Maximizing area with given perimeter constraints.

Example: A ball thrown upward:

Linear Functions and Regression

Linear Regression and Best-Fit Line

Linear regression finds the line that best fits a set of data points. The equation is , where m is the slope and b is the y-intercept.

• Finding the Slope:

• Using Data Points: Substitute values to solve for m and b.

Example: Given points (0, 10), (2, 18), (4, 26), (6, 34), find the regression line.

Factoring Polynomials

Factoring Techniques

Factoring is the process of expressing a polynomial as a product of simpler polynomials.

• Common Factor: Factor out the greatest common factor.

• Quadratic Factoring: Factor trinomials into two binomials.

• Special Products: Recognize patterns such as .

Example: Factor as .

Radical Equations

Solving Radical Equations

Radical equations contain variables inside a root. To solve, isolate the radical and raise both sides to the appropriate power.

• Isolate the Radical: Move all terms except the radical to the other side.

• Eliminate the Radical: Raise both sides to the power that matches the root.

• Check for Extraneous Solutions: Substitute solutions back into the original equation.

Example: Solve by squaring both sides: .

Rational Functions and Equations

Domain of Rational Functions

The domain of a rational function is all real numbers except where the denominator is zero.

• Set Denominator ≠ 0: Solve for values that make the denominator zero and exclude them.

Example: For , the domain is all real numbers except and .

Solving Rational Equations

To solve rational equations, find a common denominator, multiply both sides, and solve the resulting equation.

• Clear Denominators: Multiply both sides by the least common denominator (LCD).

• Check Solutions: Ensure solutions do not make any denominator zero.

Example: Solve by multiplying both sides by .

Summary Table: Factoring Methods

Additional info:

• Some questions reference using calculators for graphing and regression, which is standard in College Algebra.

• Projectile motion and area optimization are classic applications of quadratic functions.

• Factoring, radical, and rational equations are core topics in College Algebra.