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Essential Concepts

Introduction to Quadratic Functions and Parabolas

• A quadratic function is a second-degree polynomial function
• The direction of the parabola depends on the sign of
  • If [latex]a > 0[/latex], parabola opens upward
  • If [latex]a < 0[/latex], parabola opens downward
• Every parabola has an axis of symmetry that passes through the vertex
• Domain of all quadratic functions is all real numbers
• Range depends on direction of opening and vertex location
• The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
• The axis of symmetry is the vertical line passing through the vertex.
• Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.
• Quadratic functions of form [latex]f(x)=ax^2+bx+c[/latex] may be graphed by evaluating the function at various values of the input variable [latex]x[/latex] to find each coordinating output [latex]f(x)[/latex]. Plot enough points to obtain the shape of the graph, then draw a smooth curve between them.
• The vertex can be found from an equation representing a quadratic function.
• The vertex (the turning point) of the graph of a parabola may be obtained using the formula [latex]\left( -\dfrac{b}{2a}, f\left(-\dfrac{b}{2a}\right)\right)[/latex]
• The graph of a quadratic function opens up if the leading coefficient [latex]a[/latex] is positive, and opens down if [latex]a[/latex] is negative.
• Standard or vertex form, [latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex], is useful to easily identify the vertex of a parabola, where [latex]\left(h,\text{ }k\right)[/latex] is the vertex.

Complex Numbers and Operations

• The square root of any negative number can be written as a multiple of [latex]i[/latex].
• To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.
• Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.
• Complex numbers can be multiplied and divided.
• To multiply complex numbers, distribute just as with polynomials.
• To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.
• The powers of [latex]i[/latex] are cyclic, repeating every fourth one.
• A quadratic equation can have real or complex roots
• The quadratic formula provides all solutions to a quadratic equation
• The discriminant determines the nature and number of solutions
• Complex roots always come in conjugate pairs
• When a parabola doesn’t cross the [latex]x[/latex]-axis, it has complex roots

Application of Quadratic Functions

• The zeros, or [latex]x[/latex]-intercepts, are the points at which the parabola crosses the [latex]x[/latex]-axis. The [latex]y[/latex]-intercept is the point at which the parabola crosses the [latex]y[/latex]–axis.
• The vertex can be found from an equation representing a quadratic function.
• A quadratic function’s minimum or maximum value is given by the [latex]y[/latex]-value of the vertex.
• The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.
• The vertex and the intercepts can be identified and interpreted to solve real-world problems.

Key Equations

Type	Equation
difference of squares formula	[latex]a^2 - b^2 = (a+b)(a-b)[/latex]
general form of a quadratic function	[latex]f\left(x\right)=a{x}^{2}+bx+c[/latex]
standard form of a quadratic function	[latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex]
factor by grouping	[latex]ax^2 + bx + c = (px + q)(rx + s)[/latex] where [latex]pr = a[/latex] and [latex]qs = c[/latex]
discriminant	[latex]b^2 - 4ac[/latex]
discriminant Cases	[latex]b^2 - 4ac > 0[/latex]: Two real solutions [latex]b^2 - 4ac = 0[/latex]: One real solution [latex]b^2 - 4ac < 0[/latex]: Two complex solutions

Glossary

axis of symmetry

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

complex conjugate

the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number

complex number

the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[/latex], where [latex]a[/latex] is the real part, and [latex]bi[/latex] is the imaginary part

complex plane

a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number

complex roots

Solutions to a quadratic equation that contain imaginary numbers, occurring when the discriminant is negative

conjugate pair

Two complex numbers in the form [latex]a + bi[/latex] and [latex]a - bi[/latex] that occur as solutions to quadratic equations with complex roots

discriminant
the value under the radical in the quadratic formula, [latex]b^2-4ac[/latex], which tells whether the quadratic has real or complex roots

general form of a quadratic function

the function that describes a parabola, written in the form [latex]f\left(x\right)=a{x}^{2}+bx+c[/latex], where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers and [latex]a\ne 0[/latex].

imaginary number

a number in the form [latex]bi[/latex] where [latex]i=\sqrt{-1}[/latex]

standard form of a quadratic function

the function that describes a parabola, written in the form [latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex], where [latex]\left(h,\text{ }k\right)[/latex] is the vertex.

parabola

The U-shaped graph of a quadratic function

quadratic function

A polynomial function of degree 2

vertex

The highest or lowest point of a parabola [latex](h,k)[/latex]