Glossary

5.1 Quadratic Equations and Functions

4 min read•july 31, 2024

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

Top images from around the web for Characteristics of quadratic functions

  • Understand how the graph of a parabola is related to its quadratic function | College Algebra View original

    Is this image relevant?

  • Graph Quadratic Functions | Intermediate Algebra View original

    Is this image relevant?

  • Understand how the graph of a parabola is related to its quadratic function | College Algebra View original

    Is this image relevant?

1 of 3

Top images from around the web for Characteristics of quadratic functions

  • Understand how the graph of a parabola is related to its quadratic function | College Algebra View original

    Is this image relevant?

  • Graph Quadratic Functions | Intermediate Algebra View original

    Is this image relevant?

  • Understand how the graph of a parabola is related to its quadratic function | College Algebra View original

    Is this image relevant?

1 of 3
  • Quadratic function is a polynomial function of degree 2
    • Highest exponent of the variable is 2
    • : f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are real numbers and a≠0a ≠ 0
  • Graph of a quadratic function is a
    • Symmetrical, U-shaped curve

Components of a parabola

  • is the point where the parabola changes direction
    • Maximum or minimum point
    • Found using the formula x=−b/(2a)x = -b/(2a), where aa and bb are coefficients of the quadratic function in standard form
  • 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)x = -b/(2a)
  • Direction of opening determined by the sign of the leading coefficient aa in standard form
    • If a>0a > 0, parabola opens upward
    • If a<0a < 0, parabola opens downward
  • found by substituting x=0x = 0 into the function
  • x-intercepts (if they exist) found by setting the function equal to zero and solving for xx

Solving quadratic equations

Factoring method

  • Quadratic equation written in standard form: ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are real numbers and a≠0a ≠ 0
  • 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 acac and whose sum is bb
    2. Rewrite the expression as (ax+m)(ax+n)(ax + m)(ax + n), where mm and nn are the numbers found

Completing the square method

  • Solving quadratic equations by rewriting in the form (x+p)2=q(x + p)^2 = q, where pp and qq 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 xx to both sides
    3. Factor the left side as a perfect square trinomial

Quadratic formula

  • Formula used to solve any quadratic equation: x=[−b±√(b2−4ac)]/2ax = [-b ± √(b^2 - 4ac)] / 2a
    • aa, bb, and cc are coefficients of the quadratic equation in standard form
  • of a quadratic equation: b2−4acb^2 - 4ac
    • Determines the number and type of solutions ()
    • 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 (), , and dilations to the parent function f(x)=x2f(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)=ax2+bx+c+kf(x) = ax^2 + bx + c + k, where kk is the vertical shift
    • If k>0k > 0, graph shifts up; if k<0k < 0, graph shifts down
  • Horizontal shift (translation) occurs when a constant is added to or subtracted from the input (xx) of the function
    • Equation of a horizontally shifted quadratic function: f(x)=a(x−h)2+bx+cf(x) = a(x - h)^2 + bx + c, where hh is the horizontal shift
    • If h>0h > 0, graph shifts left; if h<0h < 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)=−ax2−bx−cf(x) = -ax^2 - bx - c
  • Dilation (stretch or compression) occurs when the function is multiplied by a constant factor
    • Changes the and steepness of the parabola
    • Equation of a dilated quadratic function: f(x)=ka(x−h)2+bx+cf(x) = ka(x - h)^2 + bx + c
    • If ∣k∣>1|k| > 1, results in a vertical stretch; if 0<∣k∣<10 < |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 (xx-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)=ax2+bx+cf(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 (yy-values) for the given domain
  • Range is determined by the direction of opening and the vertex of the parabola
    • If parabola opens upward (a>0a > 0), range is [vertexy,+∞)[vertex_y, +∞), meaning all yy-values greater than or equal to the yy-coordinate of the vertex
    • If parabola opens downward (a<0a < 0), range is (−∞,vertexy](-∞, vertex_y], meaning all yy-values less than or equal to the yy-coordinate of the vertex
  • In real-world contexts, interpretation of domain and range depends on quantities represented by variables xx and yy
    • Example: if xx represents time and yy represents height, domain would be limited to non-negative values, and range would represent possible heights achieved by the object at different times

Key Terms to Review (26)

Area optimization: Area optimization refers to the process of determining the dimensions that maximize or minimize the area of a given shape or region. This concept is crucial in various real-life applications, where maximizing space efficiency is essential, such as in architecture, agriculture, and packaging. It often involves using mathematical tools like calculus and quadratic equations to find optimal solutions.
Axis of Symmetry: The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. It plays a crucial role in understanding the properties of quadratic functions, as it helps locate the vertex and determine the direction of the parabola's opening. This line is significant when solving quadratic equations and analyzing systems involving parabolas, as it indicates where the function's values are symmetrical around a central point.
Completing the Square: Completing the square is a method used to convert a quadratic expression into a perfect square trinomial, making it easier to solve quadratic equations or graph quadratic functions. This technique reveals the vertex of the parabola and simplifies the process of determining the properties of quadratic equations, allowing for easy manipulation in various contexts such as transformations and geometric interpretations involving parabolas and circles.
Direction of the vertex: The direction of the vertex refers to the orientation of the vertex point in a quadratic function, which determines whether the parabola opens upward or downward. This feature is crucial for understanding the behavior of the graph, as it indicates the minimum or maximum value of the quadratic function, which can significantly affect the solutions to equations and the overall shape of the graph.
Discriminant: The discriminant is a key component of the quadratic formula, represented as $$D = b^2 - 4ac$$, which helps determine the nature of the roots of a quadratic equation. It indicates whether the roots are real and distinct, real and repeated, or complex. Understanding the discriminant is essential for solving quadratic equations and analyzing their graphs, as it directly relates to the behavior of the parabola formed by these equations.
Factored Form: Factored form refers to the expression of a polynomial as a product of its factors. This representation is crucial as it reveals the roots or x-intercepts of a polynomial function and makes it easier to analyze the behavior of the function, such as determining maximum and minimum values. Factored form is particularly important when working with quadratic equations, systems involving quadratics, and polynomial functions as it simplifies finding solutions and graphing.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions or numbers. This technique is essential for simplifying algebraic expressions, solving equations, and understanding the behavior of functions, especially polynomial and rational functions.
Graphing Solution Sets: Graphing solution sets refers to the visual representation of all possible solutions to a given mathematical problem, particularly in the context of equations and inequalities. This technique helps in understanding the relationship between variables, especially when dealing with quadratic equations, where the solution sets can be represented as parabolas on a coordinate plane. By graphing these solutions, one can easily identify key features such as intercepts, vertex, and the direction of opening, which are essential for analyzing the behavior of quadratic functions.
Maximum Value: The maximum value is the highest point on a graph of a quadratic function, representing the largest output (y-value) for a given input (x-value). This peak occurs at the vertex of the parabola when the quadratic opens downward. Understanding the maximum value is crucial because it helps identify the optimal solutions in various real-world situations, such as profit maximization in business or finding the highest point in projectile motion.
Minimum Value: The minimum value of a function is the lowest point on its graph, where the function attains its smallest output. In the context of quadratic equations, this value is particularly important as it indicates the vertex of a parabola that opens upwards. The minimum value helps to determine the range of a quadratic function and is critical for solving real-world problems that can be modeled with quadratic relationships.
Opening Direction: Opening direction refers to the way a quadratic function or parabola extends upward or downward from its vertex. This direction is determined by the sign of the leading coefficient in the quadratic equation, affecting the overall shape and behavior of the graph, including its maximum or minimum points.
Parabola: A parabola is a symmetric curve formed by the graph of a quadratic function, typically represented in the form $$y = ax^2 + bx + c$$ where 'a' determines the direction and width of the opening. It has unique properties such as a vertex, which is the highest or lowest point of the curve, and an axis of symmetry, which divides the parabola into two mirror-image halves. Understanding parabolas is essential in various mathematical contexts, including how they relate to quadratic equations, systems involving multiple equations, and their geometric characteristics when compared to other conic sections.
Projectile Motion: Projectile motion refers to the motion of an object that is thrown into the air and is subject to the force of gravity. This type of motion follows a curved path known as a parabola, influenced by both horizontal and vertical components of movement. Understanding projectile motion involves recognizing how these components interact under the influence of gravitational forces, which leads to various applications in fields like physics and engineering.
Quadratic formula: The quadratic formula is a method for solving quadratic equations of the form $$ax^2 + bx + c = 0$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a$$ is not zero. It is given by the expression $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. This formula allows us to find the roots of any quadratic equation, providing insights into the behavior of quadratic functions, their graphs, and their intersections with other equations.
Reflections: Reflections are transformations that flip a figure over a specific line, known as the line of reflection, creating a mirror image. This concept is essential in understanding geometric transformations, especially when analyzing the properties of quadratic equations and functions, as they can display symmetrical properties when graphed. Reflections can reveal important characteristics about the vertex, axis of symmetry, and the behavior of quadratic functions.
Roots: Roots refer to the values of a variable that satisfy an equation, particularly in the context of polynomial functions. They are the points at which a polynomial function equals zero, and understanding them is crucial for analyzing the behavior of functions and their graphs. The roots can provide insights into the solutions of equations and how they relate to factors and intercepts.
Solution set: A solution set is the collection of all values that satisfy a given mathematical equation or inequality. It encompasses any and all solutions that make the original statement true, providing a complete picture of possible outcomes. Understanding solution sets is crucial as they can represent ranges of numbers, specific points, or even entire functions, depending on the context of the problem being solved.
Standard Form: Standard form is a way of writing numbers or equations that allows for a clear and consistent representation. In the context of algebra, it refers to specific formats for linear equations, quadratic equations, and polynomial functions that help in solving problems and graphing them accurately.
Stretching and Compressing: Stretching and compressing refer to transformations applied to functions that alter their vertical and horizontal scales. In the context of quadratic functions, stretching means that the graph is pulled away from the x-axis, making it narrower, while compressing means that the graph is pushed closer to the x-axis, making it wider. Understanding these transformations is crucial for analyzing the behavior and shape of quadratic graphs, which can provide insights into their roots, vertex, and overall characteristics.
Test Point: A test point is a specific coordinate used to determine whether a particular region in a coordinate plane satisfies the conditions of an inequality or a function. By substituting the coordinates of a test point into an equation or inequality, one can ascertain if the point lies within a designated area, helping to visualize solutions and constraints effectively.
Translations: Translations refer to the process of shifting a graph of a function or geometric figure from one location to another without altering its shape or size. This transformation involves moving the figure in a specific direction along the x-axis and/or y-axis, and it is characterized by adding or subtracting values from the function's equation. Understanding translations is essential for grasping how changes in equations can affect the position of graphs, particularly for quadratic functions and conic sections.
Vertex: The vertex is the highest or lowest point on a parabola, depending on its orientation. It serves as a key reference point in the study of quadratic functions, indicating the maximum or minimum value of the function, and plays a crucial role in understanding the shape and position of a parabola within a coordinate system.
Vertex form: Vertex form is a way of expressing a quadratic function that highlights the vertex, or the highest or lowest point of the parabola. The standard vertex form of a quadratic equation is given as $$y = a(x-h)^2 + k$$, where (h, k) represents the coordinates of the vertex and 'a' determines the direction and width of the parabola. This form makes it easier to graph quadratic functions and understand their properties, such as the vertex location and the axis of symmetry.
Width: In the context of quadratic equations and functions, width refers to the extent of the parabola's opening, affecting how wide or narrow the parabola appears on a graph. This feature is influenced by the coefficient of the quadratic term, determining the rate at which the parabola rises or falls as it moves away from its vertex. A wider parabola indicates a smaller coefficient in front of the squared term, while a narrower parabola results from a larger coefficient.
X-intercept: An x-intercept is the point where a graph crosses the x-axis, which occurs when the value of y equals zero. This point is important for understanding the behavior of a function, as it reveals the values of x that make the function equal to zero. In different contexts, such as quadratic and polynomial functions, x-intercepts help identify roots or solutions and are crucial for graphing these types of functions effectively.
Y-intercept: The y-intercept is the point where a graph crosses the y-axis, indicating the value of the dependent variable when the independent variable is zero. This point is crucial in understanding various types of functions and their behaviors, as it helps establish the relationship between the variables at a specific point. The y-intercept also provides insights into the initial conditions of a situation represented by an equation, revealing vital information about its starting value.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide