10.5 Graphing Quadratic Equations in Two Variables

Parabolas are U-shaped curves that play a crucial role in understanding quadratic equations. They have key features like the vertex, axis of symmetry, and intercepts, which help us analyze their shape and behavior.

Graphing parabolas involves finding these key points and connecting them smoothly. This skill is essential for solving real-world problems in physics, engineering, and economics, where quadratic functions model various situations like projectile motion and profit maximization.

Understanding Parabolas and Their Key Features

Features of parabolas

• U-shaped curves symmetrical about a vertical line called the axis of symmetry
• Can open upward (concave up) or downward (concave down)
• Orientation determined by the sign of the leading coefficient $a$ in the quadratic equation $y = ax^2 + bx + c$
  • $a > 0$: parabola opens upward (positive quadratic term dominates)
  • $a < 0$: parabola opens downward (negative quadratic term dominates)
• Vertex is the turning point of the parabola, either the minimum (opens upward) or maximum (opens downward) point

Vertex and axis of symmetry

• Vertex is the turning point of the parabola, either the minimum (opens upward) or maximum (opens downward) point
  • For a quadratic equation in standard form $y = ax^2 + bx + c$, the x-coordinate of the vertex is $x = -\frac{b}{2a}$
  • y-coordinate of the vertex found by substituting the x-coordinate into the original equation
• Axis of symmetry is a vertical line that passes through the vertex
  • Equation is $x = -\frac{b}{2a}$, the same as the x-coordinate of the vertex
  • Divides the parabola into two mirror images

Intercepts of parabolas

• x-intercepts are the points where the parabola crosses the x-axis (y = 0)
  • To find x-intercepts, set $y = 0$ and solve the resulting quadratic equation
  • Solutions, if any, are the x-coordinates of the x-intercepts (can have 0, 1, or 2 x-intercepts)
  • Also known as roots of the quadratic equation
• y-intercept is the point where the parabola crosses the y-axis (x = 0)
  • To find y-intercept, set $x = 0$ in the quadratic equation and solve for $y$
  • Always has exactly one y-intercept (unless the parabola is a vertical line)

Graphing quadratic equations

• To graph a quadratic equation:
  1. Find the vertex using $x = -\frac{b}{2a}$ and calculate the corresponding y-coordinate
  2. Determine the y-intercept by setting $x = 0$ and solving for $y$
  3. Find x-intercepts, if any, by setting $y = 0$ and solving the resulting quadratic equation
  4. Plot the vertex, y-intercept, and x-intercepts (if any) on the coordinate plane
  5. Connect the points with a smooth U-shaped curve, considering the parabola's orientation based on the sign of $a$
• Parabola is symmetrical about the axis of symmetry, so points on either side are mirror images

Applications of quadratic graphs

• Quadratic equations model various real-world situations:
  • Projectile motion (height of a ball thrown upward)
  • Area optimization (dimensions of a rectangle with the largest area for a given perimeter)
  • Profit maximization (production level that yields the highest profit)
• To solve these problems:
  1. Identify the quadratic equation that models the situation
  2. Find the vertex of the parabola, representing the maximum or minimum value
  3. Interpret the vertex in the context of the problem (maximum height reached, dimensions for maximum area, production level for maximum profit)
• Understanding the properties of quadratic functions allows for optimization and problem-solving in various fields (physics, engineering, economics)

Quadratic Functions and Polynomials

• Quadratic equations are a type of polynomial function of degree 2
• Functions are relations where each input (x-value) corresponds to exactly one output (y-value)
• Polynomials are expressions consisting of variables and coefficients, using only addition, subtraction, and multiplication
• Graphing quadratic functions on a coordinate plane helps visualize their behavior across different quadrants