5.1 Quadratic Equations and Functions

Quadratic equations and functions are all about those x² terms. They're the backbone of parabolas, those U-shaped curves you've probably seen before. Understanding these equations helps you solve real-world problems involving motion, area, and more.

In this part of the chapter, we'll dive into the key features of quadratic functions, learn different ways to solve quadratic equations, and explore how to graph them. We'll also look at how these functions behave and what their domain and range mean in practical terms.

Key features of quadratic functions

Characteristics of quadratic functions

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• Quadratic function is a polynomial function of degree 2
  • Highest exponent of the variable is 2
  • Standard form: $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a ≠ 0$
• Graph of a quadratic function is a parabola
  • Symmetrical, U-shaped curve

Components of a parabola

• Vertex is the point where the parabola changes direction
  • Maximum or minimum point
  • Found using the formula $x = -b/(2a)$, where $a$ and $b$ are coefficients of the quadratic function in standard form
• Axis of symmetry is the vertical line that passes through the vertex
  • Divides the parabola into two equal halves
  • Equation of the axis of symmetry: $x = -b/(2a)$
• Direction of opening determined by the sign of the leading coefficient $a$ in standard form
  • If $a > 0$, parabola opens upward
  • If $a < 0$, parabola opens downward
• y-intercept found by substituting $x = 0$ into the function
• x-intercepts (if they exist) found by setting the function equal to zero and solving for $x$

Solving quadratic equations

Factoring method

• Quadratic equation written in standard form: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a ≠ 0$
• Factoring rewrites the equation as a product of linear factors
  • Zero-product property: if the product of two factors is zero, then at least one factor must be zero
• To factor a quadratic expression:
  1. Find two numbers whose product is $ac$ and whose sum is $b$
  2. Rewrite the expression as $(ax + m)(ax + n)$, where $m$ and $n$ are the numbers found

Completing the square method

• Solving quadratic equations by rewriting in the form $(x + p)^2 = q$, where $p$ and $q$ are constants
• To complete the square:
  1. Move the constant term to the right side of the equation
  2. Add the square of half the coefficient of $x$ to both sides
  3. Factor the left side as a perfect square trinomial

Quadratic formula

• Formula used to solve any quadratic equation: $x = [-b ± √(b^2 - 4ac)] / 2a$
  • $a$, $b$, and $c$ are coefficients of the quadratic equation in standard form
• Discriminant of a quadratic equation: $b^2 - 4ac$
  • Determines the number and type of solutions (roots)
  • If discriminant is positive, equation has two distinct real solutions
  • If discriminant is zero, equation has one repeated real solution
  • If discriminant is negative, equation has no real solutions (two complex solutions)

Graphing quadratic functions

Transformations of quadratic functions

• Graph of a quadratic function can be transformed by applying shifts (translations), reflections, and dilations to the parent function $f(x) = x^2$
• Vertical shift (translation) occurs when a constant is added to or subtracted from the function
  • Changes the y-intercept
  • Equation of a vertically shifted quadratic function: $f(x) = ax^2 + bx + c + k$, where $k$ is the vertical shift
  • If $k > 0$, graph shifts up; if $k < 0$, graph shifts down
• Horizontal shift (translation) occurs when a constant is added to or subtracted from the input ($x$) of the function
  • Equation of a horizontally shifted quadratic function: $f(x) = a(x - h)^2 + bx + c$, where $h$ is the horizontal shift
  • If $h > 0$, graph shifts left; if $h < 0$, graph shifts right

Reflections and dilations

• Reflection across the x-axis occurs when the function is multiplied by -1
  • Changes the direction of opening
  • Equation of a reflected quadratic function: $f(x) = -ax^2 - bx - c$
• Dilation (stretch or compression) occurs when the function is multiplied by a constant factor
  • Changes the width and steepness of the parabola
  • Equation of a dilated quadratic function: $f(x) = ka(x - h)^2 + bx + c$
  • If $|k| > 1$, results in a vertical stretch; if $0 < |k| < 1$, results in a vertical compression

Domain and range of quadratic functions

Domain of quadratic functions

• Domain is the set of all possible input values ($x$-values) for which the function is defined
• In real-world applications, domain may be limited by physical constraints
• For a quadratic function in standard form, $f(x) = ax^2 + bx + c$, the domain is typically all real numbers
  • Unless the context of the problem restricts the domain

Range of quadratic functions

• Range is the set of all possible output values ($y$-values) for the given domain
• Range is determined by the direction of opening and the vertex of the parabola
  • If parabola opens upward ($a > 0$), range is $[vertex_y, +∞)$, meaning all $y$-values greater than or equal to the $y$-coordinate of the vertex
  • If parabola opens downward ($a < 0$), range is $(-∞, vertex_y]$, meaning all $y$-values less than or equal to the $y$-coordinate of the vertex
• In real-world contexts, interpretation of domain and range depends on quantities represented by variables $x$ and $y$
  • Example: if $x$ represents time and $y$ represents height, domain would be limited to non-negative values, and range would represent possible heights achieved by the object at different times