Quadratic Functions

Standard Form and Key Properties

Quadratic functions are fundamental in algebra and are commonly written in the standard form:

• Standard Form:

• Direction of Opening: If , the parabola opens upward (minimum point); if , it opens downward (maximum point).

Vertex of a Quadratic Function

The vertex represents the maximum or minimum point of the parabola, depending on the sign of .

• Vertex x-coordinate:

• Vertex y-coordinate: Substitute the x-value into :

x-Intercepts and y-Intercept

• x-Intercepts: Set and solve for using factoring or the quadratic formula:

• Quadratic Formula:

• y-Intercept: Evaluate ; the y-intercept is always at .

Example: Analysis of

• Vertex: , ; Vertex is .

• x-Intercepts: Factor to get , so and ; x-intercepts are and .

• y-Intercept: ; y-intercept is .

• Maximum Value: Since , the parabola opens downward; maximum value is $8$ at the vertex.

Projectile Motion Applications

Quadratic functions model projectile motion, where the height at time is given by:

• Projectile Formula:

• Variables: = initial velocity, = initial height, = time (seconds)

Example: Ball Thrown Upward

• Given:

• Maximum Height Time: seconds

• Maximum Height: feet

Systems of Linear Equations

Solving by Elimination Method

Systems of linear equations can be solved using the elimination method, which involves combining equations to eliminate one variable.

• Example: Solve the system:

• Steps:

  1. Multiply first equation by 2:

  2. Multiply second equation by 3:

  3. Subtract:

  4. Substitute into first equation:

• Solution:

Systems with Fractions

• Example: Solve the system:

• Steps:

  1. Multiply second equation by 2:

  2. Add to first equation:

  3. Substitute into first equation:

• Solution:

Applied Word Problem: Ticket Sales

Word problems involving systems of equations often require translating real-world scenarios into algebraic equations.

• Example: A theater sold 1,095 tickets. Children's tickets sold were four times the number of adult tickets.

• Let: = children's tickets, = adult tickets

• Equations:

• Solution:

  1. Substitute into total:

• Children's tickets sold: $876$

Summary Table: Quadratic Function Properties