Quadratic Functions, Inequalities, and Zeros of Polynomial Functions – Precalculus Study Notes

Quadratic Functions and Inequalities

Converting from Standard to Vertex Form

Quadratic functions can be expressed in standard form $y = ax^2 + bx + c$ or vertex form $y = a(x-h)^2 + k$. Converting between these forms helps in graphing and analyzing the function's properties.

• Standard Form: $y = ax^2 + bx + c$

• Vertex Form: $y = a(x-h)^2 + k$

• Completing the Square: Used to convert standard form to vertex form.

Example:

• Given $y = 2x^2 - 20x + 45$

• Factor out $a$ from the first two terms: $y = 2(x^2 - 10x) + 45$

• Complete the square inside the parentheses: $x^2 - 10x + 25 - 25$

• Rewrite: $y = 2(x-5)^2 - 5$

Additional info: Completing the square is a key algebraic technique for rewriting quadratics.

Graphing Quadratic Functions

Quadratic functions graph as parabolas. The vertex form makes it easy to identify the vertex and transformations.

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

• Direction: If $a > 0$, opens upward; if $a < 0$, opens downward.

• Transformations: $a$ affects vertical stretch/compression and reflection; $h$ shifts horizontally; $k$ shifts vertically.

Example:

• $y = 2(x-2)^2 + 5$ is a parabola opening upward, vertex at $(2, 5)$, vertically stretched by 2.

• $y = -2(x+3)^2 - 7$ opens downward, vertex at $(-3, -7)$, vertically stretched by 2 and reflected over the x-axis.

Opening, Vertex, and Axis of Symmetry

Identifying the opening, vertex, and axis of symmetry is essential for graphing and analyzing quadratic functions.

• General Vertex Form: $y = a(x-h)^2 + k$

• Opening: $a > 0$ (up), $a < 0$ (down)

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

Example:

• For $f(x) = (x+5)^2 - 3$, opening: up, vertex: $(-5, -3)$, axis: $x = -5$

• For $y = ax^2 + bx + c$, vertex: $igg(- rac{b}{2a}, fig(- rac{b}{2a}ig)igg)$, axis: $x = - rac{b}{2a}$

Analyzing Parabolas: Vertex, Axis, Domain, Range, Intervals, and Intercepts

For each quadratic, determine key features: opening, vertex, axis of symmetry, domain, range, maximum/minimum values, intervals of increase/decrease, and intercepts.

• Domain: $(-, )$ for all quadratics

• Range: $[k, )$ if opens up; $(-, k]$ if opens down

• Maximum/Minimum: Vertex $k$ value

• Intervals: Increasing/decreasing split at vertex

• Intercepts: Find by setting $y=0$ (x-intercepts) and $x=0$ (y-intercept)

Example:

• For $y = - rac{1}{2}(x+4)^2 + 8$, opens down, vertex $(-4, 8)$, axis $x = -4$, range $(-, 8]$

• Quadratic formula for x-intercepts: $x = rac{-b 1    b^2 - 4ac}{2a}$

Solving Quadratic Inequalities

Graphical Method

Quadratic inequalities can be solved by graphing the corresponding parabola and identifying intervals where the inequality holds.

1. Move all terms to one side of the inequality.

2. Graph the related quadratic function.

3. Find roots (x-intercepts).

4. Determine intervals where the function is above/below the x-axis.

Example:

• $x^2 - 3x - 4 > 0$; roots at $x = -1, 4$; solution: $(-, -1)  (4, )$

• $x^2 - x  6  0$; roots at $x = -2, 3$; solution: $[-2, 3]$

Test Point/Sign Chart Method

Use test points in intervals defined by roots to determine where the inequality is satisfied.

1. Find roots of the quadratic.

2. Test values in each interval.

3. Write solution in interval notation.

Zeros of Polynomial Functions

Polynomial Long Division and Synthetic Division

Finding zeros of polynomials often involves dividing polynomials by linear factors using long division or synthetic division.

• Long Division: Divide as with numbers, aligning terms by degree.

• Synthetic Division: Efficient for divisors of the form $(x-c)$.

Example:

• Divide $x^3 - 5x^2 + 6x$ by $x - 3$ using synthetic division:

• Coefficients: 1, -5, 6, 0; divisor: 3

• Result: $x^2 - 2x + 3$

Finding Zeros

After division, set the resulting polynomial equal to zero and solve for $x$.

• Example: $x^2 - 2x + 3 = 0$

• Use quadratic formula: $x = rac{2     4 - 12}{2}$

Summary Table: Polynomial Division Methods

Method	When to Use	Steps
Long Division	Any divisor	Align terms, subtract multiples, repeat
Synthetic Division	Divisor $(x-c)$	Write coefficients, use $c$, add/multiply down

Additional info: Synthetic division is faster for linear divisors and helps quickly find zeros.

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