Functions
Concept of a Function
Vertical Line Test
If each vertical line intersects a graph in no more than one point, then the graph represents a function.
Finding the Difference Quotient
The quotient \(\frac{f(a + h) - f(a)}{h}\) is often called a difference quotient.
Linear Functions and Applications
Any function that can be written in the form
$$ f(x) = ax + b $$where \(a\) and \(b\) are real numbers, is called a linear function.
The linear function \(f(x) = x\) is often called the identity function. Any linear function of the form \(f(x) = ax + b\), where \(a = 0\), is called a constant function.
A property of plane geometry states that if two or more lines are perpendicular to the same line, then they are parallel lines.
Quadratic Functions
Any function that can be written in the form
$$ f(x) = ax^2 + bx + c $$where \(a\), \(b\) and \(c\) are real numbers and \(a \neq 0 \) is called a quadratic function. The graph of any quadratic function is a parabola (See Figure 8.23).
Vertical Translation of a Quadratic Function
In general, the graph of a quadratic function of the form \(f(x) = x^2 + k\) is the same as the graph of \(f(x) = x^2\), except that it is moved up or down \(|k|\) units, depending on whether \(k\) is positive or negative. We say that the graph of \(f(x) = x^2 + k\) is a vertical translation of the graph of \(f(x) = x^2\).
Stretching a Quadratic Function
In general, the graph of a quadratic function of the form \(f(x) = ax^2\) has its vertex at the origin and opens upward if \(a\) is positive and downward if \(a\) is negative. The parabola is narrower than the basic parabola if \(|a| > 1\) and wider if \(|a| > 1\).
Horizontal Translation of a Quadratic Function
In general, the graph of a quadratic function of the form \(f(x) = (x - h)^2\) is the same as the graph of \(f(x) = x^2\), except that it is moved to the right \(h\) units if \(h\) is positive or moved to the left \(|h|\) units if \(h\) is negative. We say that the graph of \(f(x) = (x - h)^2\) is a horizontal translation of the graph of \(f(x) = x^2\).
Transformation of a Quadratic Function
In general, the graph of a quadratic function of the form \(f(x) = a(x - h)^2 + k\) has its vertex at \((h, k)\) and opens upward if \(a\) is positive and downward if \(a\) is negative. The parabola is narrower than the basic parabola if \(|a| > 1\) and wider if \(|a| < 1>\).
Graphing Quadratic Functions of the Form \(f(x) = ax^2 + bx + c\)
The general approach is to change from the form \(f(x) = ax^2 + bx + c\) to the form \(f(x) = a(x - h)^2 + k\) by completing the square. The we apply what we saw about the transformation of quadratic functions to graph them.
More Quadratic Functions and Applications
In general, if we complete the square on:
$$ f(x) = ax^2 + bx + c $$we obtain:
$$ f(x) = a\left(x^2 + \frac{b}{a}x\right) + c $$$$ = a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}\right) + c - \frac{b^2}{4a} $$$$ = a\left(x + \frac{b}{2a}\right)^2 + \frac{4ac - b^2}{4a} $$Therefore the parabola associated with the function has its vertex at:
$$ \left(-\frac{b}{2a}, \frac{4ac - b^2}{4a}\right) $$and the equation of its axis of symmetry is:
$$ x = -\frac{b}{2a} $$
We now have another way of graphing quadratic functions of the form \(f(x) = ax^2 + bx + c\), as indicated by the following steps:
- Determine whether the parabola opens upward or downward.
- Find \(-\frac{b}{2a}\), which is the \(x\) coordinate of the vertex
- Find \(f(-\frac{b}{2a})\), which is the \(y\) coordinate o the vertex. You can also find it by evaluating:
- Locate another point on the parabola, and also locate its image across the axis of symmetry, which is the line with equation \(-\frac{b}{2a}\).
Transformations of Some Basic Curves
If we know the shapes of a few basic curves, then it is easy to sketch numerous variations of these curves by using the concepts of translation and reflection.
- Determine the domain of the function.
- Find the \(y\) intercept by evaluating \(f(0)\). Find the x intercept by finding the value(s) of \(x\) such that \(f(x) = 0\).
- Determine any types of symmetry that the equation possesses.
- Set up a table of ordered pairs that satisfy the equation.
- Plot the points associated with the ordered pairs and connect them with a smooth curve.
Vertical Translation
The graph of \(y = f(x) + k\) is the graph of \(y = f(x)\) shifted \(k\) units upward if \(k > 0\) or shifted \(|k|\) units downward if \(k < 0\).
Horizontal Translation
The graph of \(y = f(x - h)\) is the graph of \(y = f(x)\) shifted \(h\) units to the right if \(k > 0\) or shifted \(|k|\) units to the left if \(h < 0\).
Reflections of the Basic Curves
\(x\) Axis Reflection
The graph of \(y = -f(x)\) is the graph of \(y = f(x)\) reflected through the \(x\) axis.
\(y\) Axis Reflection
The graph of \(y = f(-x)\) is the graph of \(y = f(x)\) reflected through the \(y\) axis.
Vertical Stretching and Shrinking
Translations and reflections are called rigid transformations because the basic shape of the curve being transformed is not changed. In other words, only the positions of the graphs are changed.
Vertical Stretching and Shrinking
The graph of \(y = cf(x)\) is obtained from the graph of \(y = f(x)\) by multiplying the \(y\) coorindate for \(y = f(x)\) by \(c\). If \(|c|> 1\), the graph is said to be stretched by a factor of \(c\), and if \(0 < |c| < 1\), the graph is said to be shrunk by a factor of \(|c|\).
Combining Functions
In general, if \(f\) and \(g\) are functions, and \(D\) is the intersection of their domains, then the following definitions can be made:
Sum
$$ (f + g)(x) = f(x) + g(x) $$Difference
$$ (f - g)(x) = f(x) - g(x) $$Product
$$ (fg)(x) = f(x)g(x) $$Quotient
$$ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, g(x) \neq 0 $$Composition of Functions
Definition 8.2
The composition of functions \(f\) and \(g\) is defined by
$$ (f \circ g)(x) = f(g(x)) $$for all \(x\) is the domain of \(g\) such that \(g(x)\) is in the domain of \(f\).
The composition of functions is not a commutative operation.
Direct and Inverse Variation
Direct Variation
The statement \(y\) varies directly as \(x\) means:
$$ y = kx $$where \(k\) is a nonzero constant called the constant of variation or constant of proportionality.
Inverse Variation
The statement \(y\) varies inservely as \(x\) means:
$$ y = \frac{k}{x} $$where \(k\) is a nonzero constant.