Glossary

9.6 Graph Quadratic Functions Using Properties

3 min read•june 25, 2024

Quadratic functions are essential in algebra, forming curves called parabolas. They're defined by the equation f(x) = ax² + bx + c, where 'a' determines if the opens up or down. Understanding their shape and key points is crucial for graphing.

Graphing quadratics involves finding the , , and intercepts. These elements help sketch the parabola accurately. Quadratics have real-world applications in physics, economics, and optimization problems, making them a vital tool for problem-solving in various fields.

Graphing Quadratic Functions

Parabola shape of quadratics

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  • Quadratic functions have the general form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants and a≠0a \neq 0
    • a>0a > 0 results in a parabola that opens upward (U-shaped)
    • a<0a < 0 results in a parabola that opens downward ()
  • Graphs of quadratic functions are symmetrical U-shaped curves called parabolas
    • Examples: f(x)=x2f(x) = x^2 (standard parabola), f(x)=−2x2+4x−1f(x) = -2x^2 + 4x - 1 (downward-opening parabola)

Axis of symmetry and vertex

  • Axis of symmetry is a vertical line that divides the parabola into two equal halves
    • Equation of the axis of symmetry: x=−b2ax = -\frac{b}{2a}
  • Vertex is the point where the parabola crosses the axis of symmetry
    • x-coordinate of the vertex: −b2a-\frac{b}{2a}
    • y-coordinate of the vertex: substitute x-coordinate into the
  • Vertex represents the maximum or of the parabola
    • a>0a > 0: vertex is a minimum point (lowest point on the parabola)
    • a<0a < 0: vertex is a (highest point on the parabola)

Intercepts of quadratic functions

  • x-intercepts: points where the parabola crosses the x-axis (y=0y = 0)
    • Find x-intercepts by setting the quadratic function equal to zero and solving for xx
    • Quadratic functions may have 0 (no ), 1 (), or 2 () x-intercepts
    • The number of x-intercepts can be determined using the
  • : point where the parabola crosses the y-axis (x=0x = 0)
    • Find y-intercept by substituting x=0x = 0 into the quadratic function and solving for yy
    • y-intercept is always the constant term, cc, in the quadratic function

Graphing quadratics with key points

  1. Determine the direction of opening (upward if a>0a > 0, downward if a<0a < 0)
  2. Find the vertex using the axis of symmetry formula (x=−b2ax = -\frac{b}{2a})
  3. Locate the x-intercepts (solve ax2+bx+c=0ax^2 + bx + c = 0) and the y-intercept (x=0x = 0)
    • The can be used to find x-intercepts
  4. Plot the key points (vertex, intercepts) on the coordinate plane
  5. Use the symmetry of the parabola to sketch the curve through the key points
    • Points equidistant from the axis of symmetry have the same y-coordinate

Advanced Quadratic Concepts

  • is a method used to rewrite a quadratic equation in vertex form
  • can be applied to the standard quadratic function to shift, stretch, or reflect the parabola
  • The and are important elements in defining a parabola as a conic section

Real-world applications of quadratics

  • Quadratic functions model real-world situations involving maximum or minimum values
    • (trajectory of an object thrown or launched)
    • (maximizing revenue and minimizing costs)
    • (maximizing area given a fixed perimeter)
  • Steps to solve maximum or minimum problems using quadratic functions:
    1. Identify given information and the quantity to be maximized or minimized
    2. Define variables and express the quantity as a quadratic function
    3. Find the vertex of the parabola to determine the maximum or minimum value
    4. Interpret the result in the context of the problem
      • Example: A farmer wants to enclose a rectangular field with 100 meters of fencing. What dimensions will maximize the area of the field?

Key Terms to Review (24)

Area Optimization: Area optimization is the process of finding the maximum or minimum area of a geometric shape, given certain constraints or conditions. It is a fundamental concept in mathematics and is particularly relevant in the context of graphing quadratic functions.
Axis of Symmetry: The axis of symmetry is a line that divides a symmetric figure, such as a parabola, into two equal halves. It represents the midpoint or center of the symmetric figure, where the function changes direction from increasing to decreasing or vice versa.
Completing the Square: Completing the square is a technique used to solve quadratic equations by transforming them into a perfect square form. This method involves adding a constant to both sides of the equation to create a perfect square on one side, allowing for easier factorization and solution of the equation.
Directrix: The directrix is a fixed, straight line that, along with the focus, defines the shape and position of a conic section, such as a parabola, ellipse, or hyperbola. It serves as a reference point for the curve and is used in the mathematical equations that describe these geometric shapes.
Discriminant: The discriminant is a mathematical expression that determines the nature of the solutions to a quadratic equation. It plays a crucial role in understanding the behavior and characteristics of polynomial equations, quadratic equations, and their graphical representations.
Distinct Roots: Distinct roots refer to the unique solutions or values of a quadratic equation that satisfy the equation. When a quadratic equation has distinct roots, it means the equation has two separate, non-overlapping solutions that can be graphed as two distinct points on the parabolic curve.
Focus: Focus is the point at which rays of light or other radiation converge or from which they appear to diverge. It is the central point of attention, interest, or activity. In the context of graphing quadratic functions and parabolas, focus is a key characteristic that describes the shape and behavior of the graph.
Inverted U-Shaped: An inverted U-shaped curve is a graphical representation where a function initially increases, reaches a maximum point, and then decreases. This pattern is often observed in various contexts, including the relationship between a dependent variable and an independent variable.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It represents the scale or magnitude of the polynomial and plays a crucial role in various polynomial operations and properties.
Maximum Point: The maximum point of a quadratic function is the highest point on the graph of the function, representing the vertex of the parabola. It is the point where the function reaches its maximum value before decreasing.
Minimum Point: The minimum point of a quadratic function is the lowest point on the graph of the function, where the function changes from decreasing to increasing. It represents the point at which the function reaches its smallest value.
Parabola: A parabola is a curved, U-shaped line or surface that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola. Parabolas have many important applications in mathematics, physics, and engineering.
Profit Optimization: Profit optimization is the process of maximizing a business's profits by finding the optimal balance between revenue and costs. It involves analyzing and adjusting various factors, such as pricing, production, and resource allocation, to achieve the highest possible profit margin.
Projectile Motion: Projectile motion is the motion of an object that is launched or projected into the air and moves solely under the influence of gravity, without any other forces acting upon it. This type of motion is characterized by a parabolic trajectory and is governed by the principles of kinematics and the laws of motion.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. This formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Quadratic Function: A quadratic function is a polynomial function of degree two, where the highest exponent of the variable is two. These functions are characterized by a U-shaped graph called a parabola and are widely used in various mathematical and scientific applications.
Real Roots: Real roots are the solutions to an equation that are real numbers, as opposed to complex numbers. They represent the points where a graph of the equation intersects the x-axis. Real roots are an important concept in understanding the behavior and properties of quadratic equations and functions.
Repeated Root: A repeated root is a root of a polynomial equation that occurs more than once. In the context of graphing quadratic functions, a repeated root represents a point where the graph of the function touches or intersects the x-axis more than once, indicating that the function has a local maximum or minimum at that point.
Standard Form: The standard form of an equation is a specific way of writing the equation that provides a clear and organized structure, making it easier to analyze and work with the equation. This term is particularly relevant in the context of linear equations, quadratic equations, and other polynomial functions.
Transformations: Transformations in mathematics refer to the process of modifying or manipulating the shape, size, or position of a mathematical object, such as a function or a graph, without changing its essential properties. These transformations can be applied to various mathematical concepts, including functions and their graphs, to study their behavior and characteristics.
U-Shaped: The term 'U-shaped' refers to a specific shape or curve that resembles the letter 'U'. This shape is often observed in the context of graphing quadratic functions and parabolas, where the graph of the function forms a distinctive U-like curve.
Vertex: The vertex of a function or graph is the point where the graph changes direction, either from decreasing to increasing or from increasing to decreasing. It is the turning point of the graph and represents the maximum or minimum value of the function.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation crosses the x-axis. It represents the value of x when the function's y-value is zero, indicating the horizontal location where the graph intersects the horizontal axis.
Y-intercept: The y-intercept is the point where a line or graph intersects the y-axis, representing the value of the function when the independent variable (x) is equal to zero. It is a crucial concept in understanding the behavior and characteristics of various types of functions and their graphical representations.
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