3.1 Quadratic Functions
Introduction to Quadratic Functions
A quadratic function is formally defined as any function of the form f(x) = ax^2 + bx + c.
In this definition, a, b, and c are real numbers, with the crucial condition that a \ne 0 (if a were 0, the function would become linear, not quadratic).
The domain of any quadratic function is the set of all real numbers, denoted as (-\infty, \infty).
Graphs of Quadratic Functions: The Parabola
The graph of any quadratic function is a distinctive U-shaped curve called a parabola.
Shape (Concavity):
The parabola can open either upward or downward.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Vertex:
The vertex is the turning point of the parabola.
It represents either the lowest point (if opening upward) or the highest point (if opening downward).
Axis of Symmetry:
This is a vertical line that passes through the vertex.
It divides the parabola into two mirror-image halves.
Transformations:
Quadratic functions can be seen as transformations of the basic quadratic function y = ax^2.
These transformations lead to the standard form y = a(x - h)^2 + k.
The Standard Form of a Quadratic Function (Vertex Form)
The quadratic function f(x) = a(x - h)^2 + k, where a \ne 0, is known as the standard form or vertex form.
Vertex: The graph of f is a parabola whose vertex is the point (h, k).
Axis of Symmetry: The parabola is symmetric with respect to the vertical line x = h.
Opening Direction:
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Examples:
Example 1: For f(x) = -2(x - 3)^2 + 8:
Identify the vertex, intercepts, axis of symmetry (x = 3), domain ((-\infty, \infty)), and range.
Example 2: For f(x) = (x + 3)^2 + 1:
Identify the vertex, intercepts, axis of symmetry (x = -3), domain ((-\infty, \infty)), and range.
Quadratic Functions in General Form (f(x) = ax^2 + bx + c)
For a parabola defined by the quadratic function f(x) = ax^2 + bx + c:
x-coordinate of the Vertex: The x-coordinate of the vertex is given by the formula x = -\frac{b}{2a}.
Vertex Coordinates: The full vertex is found at (-\frac{b}{2a}, f(-\frac{b}{2a})) where f(-\frac{b}{2a}) is the y-coordinate obtained by substituting the x-coordinate back into the function.
Examples:
Example 3: For f(x) = -x^2 - 2x + 1:
Use the vertex and intercepts to sketch the graph, find the axis of symmetry, domain, and range.
Example 4: For f(x) = 3x^2 - 12x + 1:
Use the vertex and intercepts to sketch the graph, find the axis of symmetry, domain, and range.
Example 5: For y = 8x^2 - 40x - 130:
Find the vertex of the parabola. This information is then used to determine a reasonable viewing rectangle for a graphing utility.
Minimum and Maximum Values of Quadratic Functions
For any quadratic function f(x) = ax^2 + bx + c:
Condition for Minimum Value: If a > 0 (parabola opens upward), the function has a minimum value.
This minimum occurs at x = -\frac{b}{2a}.
The minimum value itself is f(-\frac{b}{2a}).
Condition for Maximum Value: If a < 0 (parabola opens downward), the function has a maximum value.
This maximum occurs at x = -\frac{b}{2a}.
The maximum value itself is f(-\frac{b}{2a}).
In both cases, x = -\frac{b}{2a} provides the location (the x-coordinate) of the minimum or maximum value.
The value of y, or f(-\frac{b}{2a}), provides the actual minimum or maximum value (the y-coordinate).
Example 6: For the quadratic function f(x) = -3x^2 + 6x - 13:
Determine Min/Max: Since a = -3 < 0, the parabola opens downward, and the function has a maximum value.
Find Value and Location: The maximum occurs at x = -\frac{6}{2(-3)} = -\frac{6}{-6} = 1. The maximum value is f(1) = -3(1)^2 + 6(1) - 13 = -3 + 6 - 13 = -10. So, the maximum value is -10 and it occurs at x = 1.
Identify Domain and Range: The domain is (-\infty, \infty). Since the maximum value is -10, the range is (-\infty, -10].
Real-World Application: Projectile Motion
Example 7: An archer's arrow follows a parabolic path, with its height, f(x), in feet, modeled by f(x) = -0.005x^2 + 2x + 5, where x is the horizontal distance in feet.
Maximum Height of the Arrow:
To find the maximum height, we need the y-coordinate of the vertex.
The x-coordinate of the vertex is x = -\frac{b}{2a} = -\frac{2}{2(-0.005)} = -\frac{2}{-0.01} = 200 feet.
The maximum height (y-coordinate) is f(200) = -0.005(200)^2 + 2(200) + 5 = -0.005(40000) + 400 + 5 = -200 + 400 + 5 = 205 feet.
The maximum height of the arrow is 205 feet.
Horizontal Distance to Hit the Ground:
The arrow hits the ground when its height f(x) = 0.
We need to solve the quadratic equation -0.005x^2 + 2x + 5 = 0.
Using the quadratic formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. For this equation, a = -0.005, b = 2, c = 5.
x = \frac{-2 \pm \sqrt{2^2 - 4(-0.005)(5)}}{2(-0.005)}
x = \frac{-2 \pm \sqrt{4 + 0.1}}{-0.01}
x = \frac{-2 \pm \sqrt{4.1}}{-0.01}
x \approx \frac{-2 \pm 2.02485}{-0.01}
Two possible solutions:
x_1 \approx \frac{-2 + 2.02485}{-0.01} = \frac{0.02485}{-0.01} = -2.485
x_2 \approx \frac{-2 - 2.02485}{-0.01} = \frac{-4.02485}{-0.01} = 402.485
Since horizontal distance cannot be negative, we take the positive value.
Rounding to the nearest foot, the arrow travels approximately 402 feet before it hits the ground.