Essential Concepts

Quadratic Functions

A parabola is the U-shaped graph of a quadratic function with these features:

• The vertex is the turning point of the graph; it represents the maximum (if opening down) or minimum (if opening up) value
• The axis of symmetry is a vertical line through the vertex where the parabola mirrors itself, given by [latex]x = -\frac{b}{2a}[/latex]
• The y-intercept is the point where parabola crosses the y-axis, all quadratics have a y-intercept
• The x-intercepts (zeros/roots) are points where parabola crosses the x-axis; values where [latex]y = 0[/latex]. Not all quadratics have x-intercepts.

Forms of Quadratic Functions

General Form: [latex]f(x) = ax^2 + bx + c[/latex]

• If [latex]a > 0[/latex], parabola opens upward
• If [latex]a < 0[/latex], parabola opens downward

Standard (Vertex) Form: [latex]f(x) = a(x - h)^2 + k[/latex]

• Vertex is at point [latex](h, k)[/latex]
• Makes it easy to identify transformations

Finding the Vertex from General Form

1. Find [latex]h = -\frac{b}{2a}[/latex]
2. Find [latex]k = f(h)[/latex]
3. Vertex is [latex](h, k)[/latex]

Transformations of Quadratic Functions

Starting with [latex]f(x) = x^2[/latex]:

• Vertical shift: [latex]f(x) = x^2 + k[/latex]
  • [latex]k > 0[/latex]: shift up [latex]k[/latex] units
  • [latex]k < 0[/latex]: shift down [latex]|k|[/latex] units
• Horizontal shift: [latex]f(x) = (x - h)^2[/latex]
  • [latex]h > 0[/latex]: shift right [latex]h[/latex] units
  • [latex]h < 0[/latex]: shift left [latex]|h|[/latex] units
  • Note: The sign in the formula is opposite to the direction
• Vertical stretch/compression: [latex]f(x) = ax^2[/latex]
  • [latex]|a| > 1[/latex]: narrower (vertical stretch)
  • [latex]0 < |a| < 1[/latex]: wider (vertical compression)
  • [latex]a < 0[/latex]: reflection across x-axis

Maximum and Minimum Values

To find the maximum or minimum value:

1. Determine if [latex]a[/latex] is positive (minimum) or negative (maximum)
2. Find the vertex [latex](h, k)[/latex]
3. The max/min value is [latex]k[/latex], occurring at [latex]x = h[/latex]

Polynomial Functions

A polynomial function has the form: [latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0[/latex]

• Coefficients [latex]a_i[/latex] are real numbers, [latex]a_n \ne 0[/latex]
• Powers are non-negative integers

The degree of a polynomial is the highest power of the variable

The leading term is the term with the highest degree and the leading coefficient is the coefficient of the leading term

End Behavior

End behavior describes what happens as [latex]x \to \infty[/latex] or [latex]x \to -\infty[/latex], determined by degree and leading coefficient:

Degree	Leading Coefficient	As [latex]x \to -\infty[/latex]	As [latex]x \to \infty[/latex]
Even	Positive	[latex]f(x) \to \infty[/latex]	[latex]f(x) \to \infty[/latex]
Even	Negative	[latex]f(x) \to -\infty[/latex]	[latex]f(x) \to -\infty[/latex]
Odd	Positive	[latex]f(x) \to -\infty[/latex]	[latex]f(x) \to \infty[/latex]
Odd	Negative	[latex]f(x) \to \infty[/latex]	[latex]f(x) \to -\infty[/latex]

Graphs of Polynomial Functions

Intercepts and Turning Points

• A polynomial of degree [latex]n[/latex] has at most [latex]n[/latex] x-intercepts
• A polynomial of degree [latex]n[/latex] has at most [latex]n - 1[/latex] turning points

Identifying Intercepts from Factored Form

For [latex]f(x) = a(x - r_1)(x - r_2) \cdots (x - r_n)[/latex]:

• x-intercepts: Set each factor equal to zero and solve; the zeros are [latex]r_1, r_2, \ldots, r_n[/latex]
• y-intercept: Substitute [latex]x = 0[/latex] and evaluate [latex]f(0)[/latex]

Multiplicity and Graph Behavior

Multiplicity is the number of times a factor appears in factored form.

At x-intercepts:

• Odd multiplicity (1, 3, 5…): Graph crosses the x-axis
  • Multiplicity 1: crosses like a line
  • Higher odd multiplicity: flattens as it crosses
• Even multiplicity (2, 4, 6…): Graph touches and bounces off the x-axis
  • Multiplicity 2: bounces like a parabola
  • Higher even multiplicity: flatter bounce

The sum of all multiplicities equals the degree of the polynomial.

Using Factoring to Find Zeros

To find x-intercepts:

1. Set [latex]f(x) = 0[/latex]
2. If not factored, factor using:
   • Greatest common factor
   • Trinomial factoring methods
3. Set each factor equal to zero and solve
4. If factoring isn’t possible, use technology

Graphing Polynomial Functions

Steps to sketch a polynomial graph:

1. Determine end behavior using degree and leading coefficient
2. Find intercepts:
   • x-intercepts: solve [latex]f(x) = 0[/latex]
   • y-intercept: evaluate [latex]f(0)[/latex]
3. Identify multiplicity at each x-intercept (does it cross or bounce?)
4. Plot intercepts and sketch local behavior based on multiplicity
5. Connect smoothly respecting end behavior
6. Add additional points if needed for accuracy

Intermediate Value Theorem

If [latex]f[/latex] is continuous on [latex][a, b][/latex], and [latex]f(a)[/latex] and [latex]f(b)[/latex] have opposite signs, then there exists at least one value [latex]c[/latex] in [latex](a, b)[/latex] where [latex]f(c) = 0[/latex].

Use: Confirms a zero exists between two points without finding its exact location.

Writing Formulas from Graphs

Factored form: [latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \cdots (x - r_n)^{p_n}[/latex]

Steps:

1. Identify x-intercepts from the graph
2. Determine multiplicities by observing whether the graph crosses (odd) or bounces (even)
3. Write the function with [latex]a[/latex] as unknown: [latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \cdots[/latex]
4. Find [latex]a[/latex] using another point on the graph (often the y-intercept)

Key Equations

General form of a quadratic function	[latex]f(x) = ax^2 + bx + c[/latex]
Standard (vertex) form of a quadratic function	[latex]f(x) = a(x - h)^2 + k[/latex]
The quadratic formula	[latex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/latex]
Axis of symmetry	[latex]x = -\frac{b}{2a}[/latex]
General form of a polynomial function	[latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0[/latex]
Factored form of a polynomial	[latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \cdots (x - r_n)^{p_n}[/latex]

Glossary

axis of symmetry

a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; defined by [latex]x = -\frac{b}{2a}[/latex]

coefficient

a nonzero real number multiplied by a variable raised to an exponent

degree

the highest power of the variable that occurs in a polynomial

end behavior

the behavior of the graph of a function as the input decreases without bound and increases without bound

general form of a quadratic function

the function that describes a parabola, written as [latex]f(x) = ax^2 + bx + c[/latex], where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers and [latex]a \ne 0[/latex]

global maximum

highest turning point on a graph; [latex]f(a)[/latex] where [latex]f(a) \ge f(x)[/latex] for all [latex]x[/latex]

global minimum

lowest turning point on a graph; [latex]f(a)[/latex] where [latex]f(a) \le f(x)[/latex] for all [latex]x[/latex]

Intermediate Value Theorem

for two numbers [latex]a[/latex] and [latex]b[/latex] in the domain of [latex]f[/latex], if [latex]a < b[/latex] and [latex]f(a) \ne f(b)[/latex], then [latex]f[/latex] takes on every value between [latex]f(a)[/latex] and [latex]f(b)[/latex]; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis

leading coefficient

the coefficient of the leading term

leading term

the term containing the highest power of the variable

local maximum/minimum

highest/lowest point on a graph in an open interval around [latex]x = a[/latex]

multiplicity

the number of times a given factor appears in the factored form of a polynomial; if a polynomial contains a factor [latex](x - h)^p[/latex], then [latex]x = h[/latex] is a zero of multiplicity [latex]p[/latex]

polynomial function

a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number (coefficient) and a variable raised to a non-negative integer power

standard form of a quadratic function

the function that describes a parabola, written as [latex]f(x) = a(x - h)^2 + k[/latex], where [latex](h, k)[/latex] is the vertex

term of a polynomial function

any [latex]a_ix^i[/latex] of a polynomial function in the form [latex]f(x) = a_nx^n + \cdots + a_2x^2 + a_1x + a_0[/latex]

turning point

the location at which the graph of a function changes direction; a point where the function changes from increasing to decreasing or decreasing to increasing

vertex

the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

zeros/roots

in a given function, the values of [latex]x[/latex] at which [latex]y = 0[/latex], also called roots