What is a function?
- if an equation is a function, each input will have only one output.
- when examining a graph, a function will pass the vertical line test, so the x values never repeat.
- outputs/range values can repeat (y), inputs/domain values cannot (x)
Example of a function: y = 2x-2
Example of an equation that is NOT a function: y²=x
y² = x is shown on the graph to the left. Notice that this graph would not pass the vertical line test, meaning that if you were to hold you pencil vertically and move it over the graph, the pencil would touch two parts of the graph at once. Therefore, this graph is not a function.
- Although all equations can be denoted by y=; f(x) specifically denotes functions.
Ordered pairs, maps, graphs, and tables
Ordered Pairs:
- (x,y)
- do not repeat x values (y values can repeat)
- you can list ordered pairs to graph (helpful for transitions)
Maps:
- label input and output and circle list
- list numbers in numerical order
- do not repeat any numbers
- draw lines from inputs to outputs with arrows to show direction
Graphs:
What is the graph of the function below missing in order to be technically correct? Click the Answer button below to check.
- label x and y axis (and line, especially if there is more than one graphed on a single coordinate plane)
- assign intervals for the x- and y- axis
- use arrows on both the x- and y- axis and to show that a part of an equations goes on forever
- the graph must pass the vertical line test in order to be a function
What is the graph of the function below missing in order to be technically correct? Click the Answer button below to check.
Table:
- list only x values in numerical order
- repeat numbers
Forms of a line
- y-intercept form: y=mx+b (where m=slope and b=y intercept)
- point-slope form: y-y1=m(x-x1) (where m=slope and point (x1,y1) is plugged into the equation)
- horizontal lines: y=# (any number; horizontal lines are always functions)
- vertical lines: x=# (any number; never a function- the line is vertical so it cannot pass the vertical line test)
- parallel lines: same slope with different y-intercept
- perpendicular lines: opposite reciprocal slopes
Slope formula: rise
run
m= y2-y1 (once you have assigned one ordered pair to be x1 and y1, make sure the other is x2 and y2- be consistent)
x2-x1
Example: find the slope of the line passing through the point (2,3) and (4,5)
m= 5-3 = 1
4-2
Using forms of a line
When given a slope and a y-intercept: use y-intercept form, plug the slope in for m and the y-intercept in for b
Example: slope = 3; y-int = 1
y=mx+b
y=3x+1 (shown on left, opposite graph)
You can now graph this line by plotting 2 on the y-axis and plotting further points using the slope (rise over run)
When given a point and a slope: use point-slope form, plug in the given slope and point, or plug point and slope into y-intercept form to graph
Example: slope = 2, passes through the point (3,3)
point-slope: y-3=2(x-3)
y-intercept: y=mx+b
3=2(3)+b
-3=b (plug b value in for y-intercept in equation)
Graph y=2x-3
When given 2 points: use points to find slope (see slope formula above for more instruction), then you can use the point and calculate slope to put the equation in point-slope form or y-intercept form (handy when graphing)
Example: (0,1) and (1,4)
Find slope:
m = 4-1 = 3
1-0
Choose ONE of the ordered pairs to plug into point-slope equation, and use the slope that you calculated:
y-1 = 3(x+0)
Choose ONE of the ordered pairs to plug into y-intercept form, and use the slope that you calculated:
1=3(0)+ b
1=b
Graph y =3x+1 (shown on graph above)
Example: slope = 3; y-int = 1
y=mx+b
y=3x+1 (shown on left, opposite graph)
You can now graph this line by plotting 2 on the y-axis and plotting further points using the slope (rise over run)
When given a point and a slope: use point-slope form, plug in the given slope and point, or plug point and slope into y-intercept form to graph
Example: slope = 2, passes through the point (3,3)
point-slope: y-3=2(x-3)
y-intercept: y=mx+b
3=2(3)+b
-3=b (plug b value in for y-intercept in equation)
Graph y=2x-3
When given 2 points: use points to find slope (see slope formula above for more instruction), then you can use the point and calculate slope to put the equation in point-slope form or y-intercept form (handy when graphing)
Example: (0,1) and (1,4)
Find slope:
m = 4-1 = 3
1-0
Choose ONE of the ordered pairs to plug into point-slope equation, and use the slope that you calculated:
y-1 = 3(x+0)
Choose ONE of the ordered pairs to plug into y-intercept form, and use the slope that you calculated:
1=3(0)+ b
1=b
Graph y =3x+1 (shown on graph above)