5.1 Quadratic Functions

Quadratic functions are U-shaped curves with unique features like vertices and symmetry. They're essential for modeling real-world scenarios, from projectile motion to profit optimization. Understanding their properties helps solve problems and find maximum or minimum values.

Graphing quadratic functions involves analyzing coefficients and using different forms like standard and vertex. Solving quadratic equations employs techniques such as factoring and the quadratic formula. These skills are crucial for tackling various applications in science, engineering, and economics.

Quadratic Functions

Features of parabolas

• Parabolas are U-shaped curves that represent quadratic functions
  • Vertex is the turning point of the parabola
    • Minimum point for parabolas opening upward (mountain shape)
    • Maximum point for parabolas opening downward (valley shape)
  • Axis of symmetry is the vertical line that passes through the vertex
    • Divides the parabola into two mirror images (left and right halves)
  • Direction of opening depends on the sign of the leading coefficient $a$ in the quadratic function $ax^2+bx+c$
    • Parabola opens upward if $a>0$ (positive coefficient)
    • Parabola opens downward if $a<0$ (negative coefficient)
  • Intercepts are points where the parabola crosses the x-axis or y-axis

Graphing quadratic functions

• Standard form of a quadratic function: $f(x) = ax^2 + bx + c$
  • $a$ determines the direction of opening and the width of the parabola
    • $|a| > 1$ results in a narrower parabola (steeper curve)
    • $|a| < 1$ results in a wider parabola (flatter curve)
  • $b$ affects the horizontal shift of the parabola
    • Positive $b$ shifts the parabola to the left (in the negative x-direction)
    • Negative $b$ shifts the parabola to the right (in the positive x-direction)
  • $c$ determines the vertical shift of the parabola
    • Positive $c$ shifts the parabola up (in the positive y-direction)
    • Negative $c$ shifts the parabola down (in the negative y-direction)
• Vertex form of a quadratic function: $f(x) = a(x - h)^2 + k$
  • $(h, k)$ represents the coordinates of the vertex
    • $h$ is the x-coordinate, determines the horizontal shift
    • $k$ is the y-coordinate, determines the vertical shift

Vertex of quadratic functions

• Vertex formula: $x = -\frac{b}{2a}$
  • Plug the x-coordinate into the original function to find the y-coordinate of the vertex
  • Vertex is the point $(h,k)$ where $h=-\frac{b}{2a}$ and $k=f(h)$
• For parabolas opening upward $(a > 0)$, the vertex represents the minimum value of the function
• For parabolas opening downward $(a < 0)$, the vertex represents the maximum value of the function

Solving quadratic equations

• Factoring: Rewrite the quadratic expression as a product of linear factors
• Completing the square: Rewrite the quadratic expression as a perfect square trinomial plus a constant
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for solving $ax^2 + bx + c = 0$

Domain and range

• Domain: All real numbers for most quadratic functions
• Range: Depends on the direction of opening and the vertex
  • For $a > 0$: $[k, \infty)$ where $k$ is the y-coordinate of the vertex
  • For $a < 0$: $(-\infty, k]$ where $k$ is the y-coordinate of the vertex

Applications of quadratic functions

1. Identify the given information and the quantity to be optimized (area, profit, cost)
2. Define variables and express the quantity as a quadratic function in terms of those variables
3. Find the vertex of the quadratic function using the vertex formula $x=-\frac{b}{2a}$
4. Interpret the vertex in the context of the problem

• x-coordinate $h$ represents the input value that optimizes the quantity (width, time)
• y-coordinate $k$ represents the optimized value of the quantity (maximum area, maximum profit)

1. Consider any constraints or limitations given in the problem (non-negative values, integer solutions)
2. Clearly state the solution to the optimization problem (optimal dimensions, maximum revenue)