Characteristics of a quadratic curve #
In this lesson we will be exploring four different characteristics of a quadratic curve.
- Concavity
- y-Intercept
- Vertex (turning point)
- Equation of the line of symmetry
Concavity #
Before delving into the concavity of a quadratic curve, lets investigate what is concavity of a function over an interval. Simply put, the rate at which the slope of a function changes is called its concavity over an interval.
Smelling second derivative, aren’t you!
The slope of a function at a given point is obtained by its first derivative and the rate of change of slope is simply its second derivative.
Lets put it in the context of a quadratic function:
\[f(x) = ax^2 + bx + c\]The first and second derivatives:
\[\begin{aligned} f'(x) &= 2ax + b \\ f''(x) &= 2a \\ \end{aligned}\]The second derivative only depends on the value of \( a \).
If \( a \) is positive, \( a > 0 \), then the curve opens up and it is called concave up and if \( a \) is negative, \( a < 0 \), the curve opens down and is called concave down.
y-Intercept #
The point where a curve intersects the \( y \)-axis is called y-intercept. By setting the value of \( x \) to \( 0 \) we can find the y-intercept.
For quadratic function:
\[\begin{aligned} f(x) &= ax^2 + bx + c \\ f(0) &= a(0)^2 + b(0) + c \\ f(0) &= c \end{aligned}\]the value of \( c \) gives the y-intercept. The point where the quadratic curve cuts the \( y \)-axis is \( (0, c) \).
Vertex #
The turning point of a curve. If a quadratic function is concave up, it will have a minimum point and if it is concave down, it will have a maximum point.
The maximum or a minimum points can be obtained by taking the first derivative of a function and setting it to \( 0 \).
First derivative of a quadratic function is:
\[\begin{aligned} f(x) &= ax^2 + bx + c \\ f'(x) &= 2ax + b \end{aligned}\]Setting it to \( 0 \) gives the value of \( x \),
\[\begin{aligned} f'(x) &= 0 \\ 2ax + b &= 0 \\ 2ax &= -b \\ x &= -\frac{b}{2a} \end{aligned}\]Solving for \( f(-\frac{b}{2a}) \) to get the \( y \)-coordinate of the turning point,
\[\begin{aligned} f(-\frac{b}{2a}) &= a(-\frac{b}{2a})^2 + b(-\frac{b}{2a}) + c \\ f(-\frac{b}{2a}) &= a(\frac{b^2}{4a^2}) - (\frac{b^2}{2a}) + c \\ f(-\frac{b}{2a}) &= \frac{b^2}{4a} - \frac{b^2}{2a} + c \end{aligned}\]Make the denominator of every fraction equal to \( 4a \) to make life easy,
\[\begin{aligned} f(-\frac{b}{2a}) &= \frac{b^2}{4a} - \frac{2b^2}{4a} + \frac{4ac}{4a} \\ f(-\frac{b}{2a}) &= \frac{b^2 - 2b^2 + 4ac}{4a} \\ f(-\frac{b}{2a}) &= \frac{-b^2 + 4ac}{4a} \\ f(-\frac{b}{2a}) &= -\frac{b^2}{4a} + c \end{aligned}\]So the turning point, vertex, of a quadratic curve is given by:
$$ (-\frac{b}{2a}, -\frac{b^2}{4a} + c) $$
Equation of the Line of Symmetry #
If you fold a curve along an axis such that one half is identical to the other half then the axis is called the line of symmetry.
A quadratic function always has a line of symmetry parallel to the \( y \)-axis and it passes through the turning point.
So the equation of the line of symmetry for a quadratic curve is given by:
$$ x = -\frac{b}{2a} $$