Gradient
What is the gradient of a line that rises 4 units for every 2 units it runs?
Gradient = rise/run = 4/2 = 2
In y = 3x + 2, what does the number 3 represent?
The gradient (m)
What does a horizontal line represent on a distance-time graph?
The object is stationary (not moving)
What is the gradient of a horizontal line?
0
A gym charges a $20 joining fee and $10 per week. Write a linear equation to represent the total cost C after w weeks.
C=10w+20
Find the gradient of the line passing through (2, 3) and (6, 7).
(7−3)/(6−2) = 4/4 = 1
What is the y-intercept in the equation y = -2x + 7?
7
On a distance-time graph, a steeper line means what about the object’s speed?
It is moving faster
What is the gradient of a vertical line?
Undefined
A car rental costs $50 per day plus a one-time insurance fee of $30. How much will it cost to rent the car for 3 days?
50×3 +30 = 150+30= $180
A line goes through the points (-3, 4) and (3, -2). What is its gradient?
(-2−4)/(3−(-3)) = -6/6 = -1
Write the equation of a line with a gradient of 4 and y-intercept of -1.
y = 4x - 1
How would you identify an object returning to the starting point on a distance-time graph?
A downward-sloping line returning to 0 distance
What is the equation of a vertical line passing through (3,0)?
x = 3
Jane earns $15 per hour at her job. If she works 8 hours each day, write an equation for her daily earnings E, and find how much she earns in a 5-day week.
E = 15×8 = 120
Weekly earnings: 120×5=$600
A line goes through the points (-4, -6) and (5, 3). What is its gradient?
Gradient:
m=3−(−6) / 5−(−4)=9 / 9=1
A line has the equation y = mx + c passes through (4, 2) and has a y-intercept of -6. Find m.
To find gradient:
2 = 4m - 6
4m = 8
m = 2
Looking at a distance-time graph, how could you tell if a car was accelerating?
An upward curved line, or a positive change in gradient (steeper line)
What is the equation of a horizontal line passing through (1,6)?
y = 6
A car travels at a constant speed. If it covers 180 km in 3 hours, write an equation for distance d in terms of time t. How far will it travel in 5 hours?
Speed: 60 km/h ⇒ d=60t
In 5 hours: d = 60×5 = 300 km
Find the gradient of the line passing through the points (7, -3) and (2, 5).
Gradient:
m=(5−(−3)) / (2−7)=−8 / 5
A line has the equation y = mx + c and passes through (3, 7) and (0, 1). Find m and c.
m = (7−1)/(3−0) = 6/3 = 2;
passes through (0,1), so c = 1 ⇒ y = 2x + 1
A person walks 5 km in 1 hour, rests for 2 hours, and then walks 5 km in another hour. Sketch and describe the distance-time graph.
Sloped line (5 km in 1 hr), horizontal line (2 hrs), another sloped line (5 km in 1 hr);
What is the equation of the x-axis?
y = 0
A mobile plan charges a base fee of $5 and $0.10 per text message. Write an equation for the total cost C of sending m messages, and find the cost of sending 250 messages.
C = 0.10m+5
C = 0.10×250+5 = 25+5 = $30