Investigation: Graphing Linear Relations
Ex 4F: Gradient-Intercept Form (y=mx+c)
Move the sliders for and to explore the properties of a straight line graph, otherwise known as a LINEAR GRAPH.
Question 1:
What feature of the red line does represent?
Question 2:
a) What do you notice about the red line as gets larger (greater than 1)?
b) What do you notice about the red line as gets smaller (between 0 and 1)?
c) What do you notice when is negative?
d) What do you notice when ?
e) What feature of the red line does represent?
Question 3:
Using your investigation from Questions 1 and 2, move the sliders for and to determine the equation of
at least 2 random lines.
(Feel free to find the equations of more than just 2 lines. Practice makes perfect!)
Visualizing the Gradient (m) of a Straight Line
In Mathematics we have a special term to refer to the ‘steepness’ / ‘slope’ of a line. We call it the GRADIENT .
The gradient of a straight line is denoted by the symbol in the linear equation: .
The definition of a gradient is the:
“Change in vertical distance as the horizontal distance increases by 1 unit”.
However, we more often use the definition/formula:
Lets investigate... Question 4: Calculate the gradient between the points: a) (0,0) and (1,2) b) (0,0) and (2,2) c) (0,0) and (2,1) d) (0,0) and (3,-3) d) (0,0) and (2,3) (Hint: Change the value of first.) e) When finding the gradient of a linear graph, which points can be used to calculate ? (Hint: The blue dot on the can be dragged up and down your line using your cursor.) Question 5: a) In the equation of a straight line, the gradient () is the coefficient of ___ b) Describe how you would determine the gradient of a straight line from its graph? c) Can the between two points be negative? Can the between two points be negative? d) What must the gradient be if the rise between two points is zero ()? e) How do you calculate the gradient of a straight line when only given the coordinates of two points?Ex 4G: Finding the Equation of a Straight Line
Put your new skills to the test! Try to determine the equation of the following 9 lines!
Question 6:
Determine the equations of the lines a) though i).
Check that each of your equations are correct by typing them in and seeing if you get a match.
(Two blue points have been given in the top-left which can be moved to help visualize your calculation of 'rise' and 'run')
4C: Special Cases - Lines with Only One Intercept
Any two points can form a a straight line. Let's investigate how we use this to sketch horizontal and vertical lines
Question 7:
Try to make vertical and horizontal lines:
a) b) c)
d) e) f)
Question 8:
a) How do you determine the equation of a vertical line given its linear graph?
b) How do you determine the equation of a horizontal line given its linear graph?
Domain and Range
You may have noticed that linear graphs can be very long (in fact they are infinitely long!). So what must we do if we only want a segment of a line? ... We use LINE SEGMENTS.
Below you can sketch a sloped / horizontal line segment and a vertical line segment.
Try entering your straight line equations into the input boxes (just as with Q6), but this time include maximum and minimum values for and !
Be sure to see what happens if you 'tick' the "Show Max/Min Lines" box during your investigation.
Question 9:
a) What does the 'domain' of a linear graph represent?
b) What does the 'range' of a linear graph represent?
c) Can you think of mnemonic device that which help you remember the differences and/or similarities of
'domain' and 'range'.
Question 10:
a) Sketch a positively sloped line segment and vertical line segment that join at end points.
b) Sketch a negatively sloped line segment and vertical line segment that join at end points.
c) Sketch a horizontal line segment and vertical line segment that join at end points.
Playground: Sketch all the lines and segments that you like!
(Coming soon...)
What is a Linear Relation? (Interpreting m and c)
(Coming soon...)
Extension: Graphical Solutions to Simultaneous Equations
(Coming soon...)
4I: Parallel and Perpendicular Lines
(Coming soon...)