Quadratic Functions

Vertex, Intercepts, and Graphing

Quadratic functions are polynomial functions of degree 2, typically written as . Understanding their key features is essential for graphing and solving related problems.

• Vertex: The vertex is the highest or lowest point on the graph. It can be found using the formula: To find the y-coordinate, substitute this x-value into the function:

• x-intercepts: Set and solve for (roots of the equation).

• y-intercept: Set and solve for .

• Graph: Plot the vertex, x-intercepts, and y-intercept, then sketch the parabola.

• Local Maximum/Minimum:

  • If , the parabola opens downward and the vertex is a maximum.

  • If , the parabola opens upward and the vertex is a minimum.

Example: For :

• , ,

• Vertex: ;

• Vertex at (local minimum since )

• x-intercepts: Solve

• y-intercept:

Example (Word Problem): The profit for selling pretzels. The maximum profit occurs at pretzels. Maximum profit: .

Polynomial Functions

Key Properties and Graphing

Polynomial functions generalize quadratics to higher degrees. Their graphs and properties depend on degree, leading coefficient, and factorization.

• Standard Form:

• End Behavior: Determined by the leading term .

  • If is even and , both ends up ( )

  • If is even and , both ends down ( )

  • If is odd and , left down, right up ( )

  • If is odd and , left up, right down ( )

• x-intercepts (Real Zeros): Set and solve for .

• y-intercept:

• Multiplicity: The exponent of each factor at a zero .

  • Even multiplicity: The graph touches the x-axis and turns around.

  • Odd multiplicity: The graph crosses the x-axis.

• Maximum Number of Turning Points: At most for degree .

• Graphing Steps:

  1. Determine end behavior from leading term.

  2. Find and plot real zeros and their multiplicities.

  3. Find and plot the y-intercept.

  4. Estimate and plot turning points (up to ).

  5. Connect points smoothly, respecting end behavior and multiplicity at zeros.

Table: Zeros, Multiplicity, and Graph Behavior

Additional info: Table inferred from context; actual zeros and multiplicities may vary by example.

Example: Factored Polynomial

Given :

• Degree:

• End behavior: Like (odd degree, positive leading coefficient): left down, right up

• Real zeros: (multiplicity 3, cross), (multiplicity 2, touch)

• Maximum turning points:

• y-intercept:

Polynomial Inequalities

Solving and Interpreting Solutions

Polynomial inequalities involve finding intervals where a polynomial is greater than or less than zero. The solution process involves analyzing the sign of the polynomial across its real zeros.

• 1. Arrange the inequality: Move all terms to one side so the other side is zero.

• 2. Find x-intercepts: Solve to find critical points (real zeros).

• 3. Determine multiplicity: At each zero, decide if the graph crosses (odd) or touches (even) the x-axis.

• 4. Plot zeros on a number line: Mark intervals between zeros.

• 5. Test intervals: Choose a test value in each interval and substitute into to determine the sign.

• 6. Assign intervals: Use the original inequality to select intervals (e.g., means where is positive).

• 7. Bracket notation:

  • Use parentheses for strict inequalities ( or ).

  • Use brackets for inclusive inequalities ( or ) at zeros with odd multiplicity.

Example: Solve

• Zeros: (multiplicity 2, touch), (multiplicity 1, cross)

• Test intervals: , ,

• Test values: , ,

• Determine sign in each interval and select intervals where

• Solution: (intervals where the polynomial is non-negative)

Additional info: The process of sign analysis and interval selection is fundamental for all polynomial inequalities.