📘Intermediate Algebra Unit 9 – Quadratic Equations and Functions

Key Concepts and Definitions

• Quadratic equation an equation that can be written in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$
• Quadratic function a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$
  • The graph of a quadratic function is a parabola
• Parabola a symmetrical, U-shaped curve that is the graph of a quadratic function
  • The axis of symmetry is the vertical line that divides the parabola into two equal halves
• Vertex the point where a parabola changes direction, either a maximum or minimum point
• Discriminant the value of $b^2 - 4ac$ in a quadratic equation, determines the number and type of solutions
  • If the discriminant is positive, there are two real solutions
  • If the discriminant is zero, there is one real solution (a double root)
  • If the discriminant is negative, there are no real solutions (two complex solutions)

Quadratic Equation Basics

• Standard form of a quadratic equation $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$
• Coefficients in a quadratic equation, $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant term
• Leading coefficient the coefficient of the highest degree term in a polynomial (in quadratic equations, the coefficient of $x^2$)
• Degree of a quadratic equation always 2, as the highest power of the variable is 2
• Quadratic equations can be solved by factoring, completing the square, using the quadratic formula, or graphing
• Solutions (roots) of a quadratic equation the values of $x$ that satisfy the equation, making it equal to zero
  • A quadratic equation can have 0, 1, or 2 real solutions, depending on the value of the discriminant

Solving Quadratic Equations

• Factoring method of solving quadratic equations by rewriting the equation as a product of linear factors
  • Steps: 1) Set the equation equal to zero, 2) Factor the left side of the equation, 3) Set each factor equal to zero and solve for $x$
• Completing the square method of solving quadratic equations by creating a perfect square trinomial
  • Steps: 1) Move the constant term to the right side of the equation, 2) Factor out the coefficient of $x^2$, 3) Add the square of half the coefficient of $x$ to both sides, 4) Factor the left side into a perfect square, 5) Take the square root of both sides and solve for $x$
• Quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of a quadratic equation in standard form
  • Can be used to solve any quadratic equation, regardless of its form or the value of the discriminant
• Graphing method of solving quadratic equations by graphing the function and finding the $x$-intercepts (points where the graph crosses the $x$-axis)
  • The $x$-coordinates of the $x$-intercepts are the solutions to the quadratic equation

Graphing Quadratic Functions

• Parabola the graph of a quadratic function, a symmetrical U-shaped curve
• Axis of symmetry the vertical line that divides a parabola into two equal halves, passes through the vertex
  • Equation of the axis of symmetry: $x = -\frac{b}{2a}$
• Vertex the point where a parabola changes direction, either a maximum or minimum point
  • Coordinates of the vertex: $(-\frac{b}{2a}, f(-\frac{b}{2a}))$
• Concavity the direction in which a parabola opens, either upward (concave up) or downward (concave down)
  • If $a > 0$, the parabola opens upward (concave up)
  • If $a < 0$, the parabola opens downward (concave down)
• $x$-intercepts the points where a parabola crosses the $x$-axis, found by setting $y = 0$ and solving for $x$
  • The $x$-coordinates of the $x$-intercepts are the solutions to the quadratic equation
• $y$-intercept the point where a parabola crosses the $y$-axis, found by setting $x = 0$ and solving for $y$
  • The $y$-coordinate of the $y$-intercept is the constant term $c$ in the quadratic function

Properties of Quadratic Functions

• Domain the set of all possible input values (x-values) for a function
  • For quadratic functions, the domain is all real numbers
• Range the set of all possible output values (y-values) for a function
  • For quadratic functions, the range depends on the concavity and the vertex
    • If the parabola opens upward (concave up), the range is $[y_v, \infty)$, where $y_v$ is the y-coordinate of the vertex
    • If the parabola opens downward (concave down), the range is $(-\infty, y_v]$, where $y_v$ is the y-coordinate of the vertex
• Extrema the maximum or minimum value of a function
  • For quadratic functions, the extremum occurs at the vertex
    • If the parabola opens upward (concave up), the vertex is a minimum point
    • If the parabola opens downward (concave down), the vertex is a maximum point
• Symmetry quadratic functions are symmetrical about the axis of symmetry
  • If $(x, y)$ is a point on the parabola, then $(-x, y)$ is also a point on the parabola
• Transformations changes to the graph of a quadratic function based on the values of $a$, $b$, and $c$
  • Vertical stretch/compression: $|a|$ determines the vertical stretch (if $|a| > 1$) or compression (if $0 < |a| < 1$)
  • Horizontal shift: $-\frac{b}{2a}$ determines the horizontal shift, positive for left and negative for right
  • Vertical shift: $c$ determines the vertical shift, positive for up and negative for down

Real-World Applications

• Projectile motion the path of an object launched at an angle to the horizontal, neglecting air resistance
  • The height of the object as a function of time is a quadratic function: $h(t) = -\frac{1}{2}gt^2 + v_0 \sin(\theta) t + h_0$, where $g$ is the acceleration due to gravity, $v_0$ is the initial velocity, $\theta$ is the launch angle, and $h_0$ is the initial height
• Optimization problems finding the maximum or minimum value of a quadratic function in a real-world context
  • Example: A farmer has 100 meters of fencing to enclose a rectangular field. What dimensions will maximize the area of the field?
• Revenue and profit functions quadratic functions that model a company's revenue or profit based on the quantity of a product sold
  • Revenue function: $R(x) = px - \frac{b}{2a}x^2$, where $p$ is the price per unit and $x$ is the quantity sold
  • Profit function: $P(x) = R(x) - C(x)$, where $R(x)$ is the revenue function and $C(x)$ is the cost function
• Acceleration and velocity the relationship between an object's acceleration, velocity, and position can be modeled with quadratic functions
  • If an object has a constant acceleration, its velocity as a function of time is a linear function, and its position as a function of time is a quadratic function

Common Mistakes and How to Avoid Them

• Forgetting to set the equation equal to zero before factoring or using the quadratic formula
  • Always start by setting the equation equal to zero to ensure proper form
• Misidentifying the coefficients $a$, $b$, and $c$ in a quadratic equation
  • Take care to correctly identify the coefficients, especially when the equation is not in standard form
• Incorrectly factoring the quadratic expression
  • Double-check your factoring by multiplying the factors to ensure they equal the original expression
• Misapplying the quadratic formula, especially with negative coefficients
  • Pay close attention to the signs of the coefficients and the order of operations when using the quadratic formula
• Graphing the parabola with incorrect concavity
  • Remember that the concavity is determined by the sign of the leading coefficient $a$
• Confusing the x-coordinate and y-coordinate of the vertex
  • The x-coordinate of the vertex is $-\frac{b}{2a}$, and the y-coordinate is found by evaluating the function at this x-value
• Misinterpreting the real-world meaning of the solutions in application problems
  • Always consider the context of the problem when interpreting the solutions, and discard any solutions that don't make sense in the given context

Practice Problems and Tips

• Practice factoring quadratic expressions with various coefficients and constant terms
  • Focus on identifying common factors, difference of squares, and perfect square trinomials
• Solve quadratic equations using multiple methods (factoring, completing the square, quadratic formula) to reinforce understanding
  • Compare the solutions obtained from different methods to ensure consistency
• Graph quadratic functions by hand, paying attention to the concavity, vertex, and intercepts
  • Use transformations to graph quadratic functions more efficiently
• Analyze the properties of quadratic functions from their equations and graphs
  • Identify the domain, range, extrema, and symmetry of the function
• Apply quadratic functions to real-world problems, such as projectile motion and optimization
  • Practice setting up the quadratic function based on the given information and interpreting the solutions in context
• Create your own practice problems by modifying the coefficients and constant terms of quadratic equations and functions
  • Observe how these changes affect the solutions and the graph of the function
• Collaborate with classmates to discuss problem-solving strategies and compare solutions
  • Explaining concepts to others can deepen your own understanding
• Seek additional help from your instructor, tutoring services, or online resources if needed
  • Don't hesitate to ask for clarification or further explanation on challenging topics