Quadratic Functions
A quadratic function is a polynomial function of degree 2, typically expressed 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 called a parabola, which is a symmetric, U-shaped curve.
Standard Form
The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. In this form:
- $a$ determines the direction and steepness of the parabola (for $a>0$ , the graph is concave up, and for $a<0$ is concave down
- $b$ influences the axis of symmetry
- $c$ is the y-intercept
Consider the function $f(x) = 2x^2 - 4x + 3$. Here, $a=2$, $b=-4$, and $c=3$.
Y-intercept
The y-intercept of a quadratic function is the point where the parabola crosses the y-axis. It occurs when $x = 0$, and its coordinates are always (0, c).
For $f(x) = 2x^2 - 4x + 3$, the y-intercept is (0, 3).
Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Its equation is given by:
$$ x = -\frac{b}{2a} $$
For $f(x) = 2x^2 - 4x + 3$, the axis of symmetry is $x = -\frac{-4}{2(2)} = 1$.
Factored Form
The factored form of a quadratic function is $f(x) = a(x-p)(x-q)$, where $p$ and $q$ are the x-intercepts (roots) of the function. This form is particularly useful for finding the zeros of the function.
The function $f(x) = 2(x-1)(x-2)$ has x-intercepts at $x=1$ and $x=2$.
Not all quadratic functions can be easily factored, especially if they have complex roots.
Vertex Form
The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where (h, k) represents the vertex of the parabola. This form is particularly useful for graphing a function.
The function $f(x) = 2(x-3)^2 - 4$ has its vertex at (3, -4).
Converting Between Forms
Students are expected to be able to convert quadratic functions between standard, factored, and vertex forms. Each conversion involves algebraic manipulation and sometimes the use of the quadratic formula.
To convert from standard to vertex form, complete the square. To convert from standard to factored form, factor the quadratic expression or use the quadratic formula if necessary.
Transformations
Quadratic functions can undergo various transformations:
- Vertical stretch/compression: Changing $a$ in $f(x) = ax^2$
- Vertical shift: Adding or subtracting a constant $k$ to get $f(x) = ax^2 + k$
- Horizontal shift: Replacing $x$ with $(x-h)$ to get $f(x) = a(x-h)^2$
Starting with $f(x) = x^2$:
- $g(x) = 2x^2$ is a vertical stretch by a factor of 2
- $h(x) = x^2 + 3$ is a vertical shift up by 3 units
- $j(x) = (x-2)^2$ is a horizontal shift right by 2 units
These questions highlight the unique characteristics of mathematical knowledge and its applications, encouraging students to think critically about the nature of mathematics as a discipline.
