functions

Graphing a linear function

A graph of an equation in two variables is the set of all points that satisfy the equation.

If you’re given the equation y = 3x-2 and asked to graph it, you would do the following steps. (Note: This equation is a linear equation, which means it will appear as a straight line). The first step is to set up a table of x and y.

y = 3x-2

​Assign values to x, then figure out what the value for y will be. For example, if x = 0, then y = -2. If x = 1 then, y = 1, and if x=2/3, then y = 0. Note that it’s good to choose x=0 and y = 0 because x intersects the y-axis when it has the value of 0, and y intersects the  x–axis when it has the value of 0.

From the table above, we have the points (0, -2), (1, 1) and (2/3, 0). However, to draw a line you would only need to plot two points and then connect those dots. Since "linear" equations produce a straight line, you might as well use your ruler for this part.

You can see in the graph below these points are connected in a straight line.

Press the "Play Button" on navigation bar to see the steps.

Definition

A linear function could be written in the following standard equation y = f(x) =  bx+ a. So, a linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y.  The highest power over the x variable  in a linear function is 1.

Example:  y = 2x-1  is a straight-line equation, where b=2 and a= -1. 

Exercise. Drag Point A and explore how line-equation changes.

• How does the line look when "b" is positive?
• How does the line look when "b" is negative?
• What is the relation between coordinates of  point "A"  and the line equation?
• What is the line equation when you place "A" on the origin"?
• What position does the line take when  the line equation changes to y=a (notice that there is no x variable in the equation)?  How does the line look (parallel to what axis)?

Explore quadratic functions

A Quadratic function comes with the standard form of y = ax² + bx + c, when the highest power over the x variable(s) is 2.

Please note that when "a" is positive, the parabola opens upward, and the vertex is the minimum value of the function. On the other hand, if "a" is negative, the graph opens downward, and the vertex is the maximum value of the function.

Exercise:

Below, explore the interactive graph that allows you to change the a, b, and c values in a quadratic equation and view how the resulting parabola changes.  Please note that the initial quadratic function shown in the graph is y = x² (refresh the graph to return to y = x²).

• How does  the parabola behave when "a" becomes positive?
• How does the parabola behave when "a" becomes negative?
• How does the parabola behave when "a" approaches 0 ?
• What happens if "a=0"?
• Describe how the parabola moves when
  1.  b = 0,
  2. b > 0  
  3. b < 0
• Describe how the parabola moves when
  1.  c = 0, 
  2. c > 0  
  3. c < 0

Definition

Taking the absolute value of a negative number makes it positive. For this reason, graphs of absolute value functions tend not to look quite like the graphs of linear functions that you've already studied.

Task:

Use the sidebars in the graph below to change the values of a, h, and k in the formula y=a|x-h|+k. Note that "a" is not inside the absolute value sign!

• Explore how the graph behaves when "a" switches from negative to positive
• Explore how the graph behaves when "h" switches from negative to positive
• Explore how the graph behaves when "k" switches from negative to positive
• Click the "random graph" button and see if you can find the values of a, h, and k that produce that graph!

Definition

The graph of the function f(x) = √x makes half a parabola . The principal square root function f(x) = √x (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself. To explore how a square root function behaves, try to change the value on the sliders.

y=a√(x-b)+c