Definition and key concepts
Definition
Linear function
A linear function is any function of the following form:
y=mx+b
m and b are real numbers.
The following function is a linear function:
y=2x-7
Graphical representation
y-intercept
The y-intercept of a function f(x) is the point (0,f(0)) in the xy-plane. Suppose that f(x)=mx+b is a linear function, then the y-intercept of the linear function is:
(0,b)
The y-intercept of the function f(x)=7x-2 is (0,-2).
The x-intercepts of a function f(x) are the collection of points in the xy-plane where the graph of f(x) touches the x-axis. A linear function has only one x-intercept:
\left(\dfrac{-b}{m},0\right)
Consider the following function:
f(x)=7x-2
Its x-intercept is:
\left(\dfrac{2}{7},0\right)
The graph of a linear function is a line through the x -intercept and y -intercept of the function.
Average rate of change and difference quotient
The average rate of change of the function f(x)=mx+b from any two values is m.
Consider the following linear function:
f\left(x\right)=3x-1
The average rate of change of f\left(x\right) from x=4 to x=5 is 3.
The average rate of change of a function f(x) from x_0 to x_1, with x_0\lt x_1, is:
\dfrac{f(x_1)-f(x_0)}{x_1-x_0}
Suppose f(x)=mx+b is a linear function. Then the average rate of change of f(x) from x_0 to x_1 is:
\frac{mx_1+b-(mx_0+b)}{x_1-x_0}=\frac{m(x_1-x_0)}{x_1-x_0}=m
Therefore the average rate of change of the function f(x)=mx+b from any two values is m.
Difference quotient
Let f\left(x\right)=mx+b be a linear function. The difference quotient of f\left(x\right) is:
\dfrac{f\left(x+h\right)-f\left(x\right)}{h}=m
The difference quotient, or the average rate of change, of a linear function is constant. This algebraic property is a reflection of the geometric property that the graph of linear function is a line through two points.
Slope
The slope of a linear function y=mx+b is the value m.
The slope of the linear function f(x)=7x-2 is 7.
The more positive or negative the slope of a linear function the more vertical the graph of a linear function is.
Particular linear functions
Constant functions
Constant function
A constant function is a function of the following form:
f(x)=b
b is a real number.
The following function is constant:
f(x)=7
If f(x)=b is a constant function, then we could also write f(x)=0x+b. Therefore a constant function is a linear function with slope 0. In particular, the graph of the constant function f(x)=b is a horizontal line with y-intercept (0,b).
By examining the graph of a constant function f(x)=b, we see that f(x) neither increases nor decreases and that all points on the graph are extrema.
Identity function and directly proportional function
Identity function
The identity function is the following linear function:
f(x)=x
It is called "identity function" since the image of a real x is x itself.
- The x-intercept and y-intercept of the identity function is (0{,}0).
- The slope of the identity function is 1.
The graph of the identity function is:
Observe that the graph the identity function is always increasing at a constant rate of 1 and that in particular the identity function has no extrema.
Directly proportional function
A directly proportional function is any linear function of the following form:
f(x)=mx
The following linear function is a directly proportional function:
f(x)=7x
If f(x)=mx is a directly proportional function, then:
- The x-intercept and y-intercept of f(x) is (0{,}0).
- The slope of f(x) is m.
The graph of f(x) is a line passing through (0{,}0) with slope m.
Vertical lines
A vertical line is represented by an equation of the form:
x=a
a is a real number.
The vertical line with equation x=a has x-intercept at (a,0).
A vertical line in the xy-plane can never be the graph of a function by the vertical line test. However, a vertical line has the property that any two points on the line have the same x-value.
Linear equations and inequalities
Linear equations
Linear equation
A linear equation is any equation that can be written as:
ax+b=0
a and b are real numbers.
The following equation is linear:
3x+2=0
Consider the following equation:
3x+2=7x-4
It is a linear equation because if we move all components of the equation to the left hand side we arrive at:
3x+2-7x+4=0
This is equivalent to the following:
-4x+6=0
A linear equation of the form ax+b=0 can be solved graphically by finding the x-intercept of the linear function y=ax+b.
We can solve the following equation graphically:
2x+1=0
The solution is:
x=-\dfrac{1}{2}
A linear equation of the form ax+b=cx+d can be solved graphically by finding the intercept of the two linear functions y=ax+b and y=cx+d.
Consider the following linear equation:
3x+2=7x-4
The solution can be found graphically by finding the intersection of the following graphs:
- y=3x+2
- y=7x-4
The two lines intercept in point \left( 1.5{,}6.5 \right). Therefore, the equation has one solution:
x=1.5
The solution of a linear equation of the form ax+b=0 is:
x=\dfrac{-b}{a}
Consider the following linear function:
3x+2=0
The solution is:
x=\dfrac{-2}{3}
The inverse of a function y=f(x) is found by solving the equation the following equation for y :
x=f(y)
Suppose y=mx+b is a linear function. Then solving x=ym+b for y is done as follows:
y=\dfrac{x-b}{m}=\dfrac{1}{m}x -\dfrac{b}{m}
Note that the inverse of a linear function is another linear function.
Consider the following function:
y=7x-2
Its inverse is:
y= \dfrac{1}{7}x+\dfrac{2}{7}
Linear inequalities
Linear inequality
A linear inequality is any inequality that can be written as one of the following forms:
- ax+b\geq 0
- ax+b>0
- ax+b\leq 0
- ax+b<0
a and b are real numbers.
The following inequality is a linear inequality:
2x-7\lt 0
The following inequality is a linear inequality:
2x-7\geq 3x+5
To prove this, move all terms to the left side of the equation to find the following:
2x-3x-7-5\geq 0
-x-12\geq 0
To solve a linear inequality we must isolate the unknown x on one side of the inequality. Much like solving a linear equation, we can move the constant term to one side of the equation and then divide by the term attached to the unknown x.
- If the term attached to x is positive, then we divide by that term to preserve the direction of the inequality.
- If the term we divide by is negative, then we must reverse the inequality.
Consider the following linear inequality:
3x+7\leq 0
It is solved as follows:
3x+7\leq 0
3x\leq -7
x\leq \dfrac{-7}{3}
Consider the following linear inequality:
-2x+7\gt 0
It is solved as follows:
-2x+7\gt 0
-2x\gt -7
It is necessary to divide by -2 to isolate x. Dividing by a negative number reverses the inequality. Therefore, the inequality becomes:
x\lt \dfrac{-7}{-2}
x\lt \dfrac{7}{2}
A linear inequality can also be solved graphically.
The following equation can be solved graphically:
3x+7\lt0
The x -values on the graph of y=3x+7 strictly below the line y=0 correspond to solutions to the inequality.
The solution set is the set of all real numbers such that x\lt -\dfrac{7}{3}.
The rules for solving a linear inequality help to solve two linear inequalities simultaneously.
Consider the following linear inequalities :
-2\leq -3x+7\lt 5
It is solved as follows:
-2\leq -3x+7\lt 5
-2-7\leq -3x \lt 5-7
-9\leq -3x\lt -2
\dfrac{-9}{-3}\geq x\gt \dfrac{-2}{-3}
3\geq x\gt \dfrac{2}{3}
The solution for the simultaneous inequality is equivalent to:
\dfrac{2}{3}\lt x\leq 3