Reading Graphs & Estimation — Challenge
A graph is an argument. Read the scale before the shape, or you will believe whatever it wants you to believe.
What these problems cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Percent — Parts per hundred. 37% = 37100 = 0.37.
- Area — Space inside, measured in square units.
- Bar graph — Compares separate categories.
- Line graph — Shows change over time.
- Circle graph — Shows parts of one whole, as percentages summing to 100%.
- Dimensions — Rows × columns. A 2 × 3 matrix has two rows and three columns.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
In which quadrant does the point (3,-2) lie?
- I
- II
- III
- IV
- On an axis
Worked answer
IV
The quadrants are numbered anticlockwise from the top right, where both coordinates are positive.
Here x is positive and y is negative, which is the bottom right.
That is quadrant IV.
Problem 2
What is the slope of the line through (1,2) and (4,11)?
- 1
- 2
- 3
- 4
- 9
Worked answer
3
Slope is the change in y divided by the change in x.
The change in y is 11-2 = 9 and the change in x is 4-1 = 3.
So the slope is 93 = 3.
Problem 3
What is the y-intercept of the line y=3x-6?
- -9
- -6
- -2
- 3
- 6
Worked answer
-6
The y-intercept is the value of y when x=0.
y = 3(0)-6 = -6.
The value 2 is the x-intercept, found by setting y=0 instead.
Problem 4
Which of these points lies on the line y=2x?
- (6,3)
- (2,5)
- (3,6)
- (1,3)
- (0,2)
Worked answer
(3,6)
A point lies on the line when its y is twice its x.
For (3,6): twice 3 is 6, so it fits.
For (6,3) the coordinates are the right numbers in the wrong order, since 3 is half of 6 rather than double.
Problem 5
What is the x-intercept of the line y=2x-8?
- -8
- -4
- 2
- 4
- 8
Worked answer
4
The x-intercept is where the line crosses the x-axis, so set y=0.
0 = 2x-8 gives 2x = 8 and x = 4.
Setting x=0 instead would give the y-intercept, -8.
Problem 6
The point (0,5) lies on which axis?
- The x-axis
- The y-axis
- Both axes
- Neither axis
- Quadrant I
Worked answer
The y-axis
A point with x=0 has not moved left or right from the origin, so it sits on the vertical axis.
That is the y-axis. Only the origin (0,0) lies on both.
Problem 7
What is the slope of a horizontal line?
- 0
- 1
- -1
- Undefined
- 12
Worked answer
0
On a horizontal line the y value never changes, so the change in y is 0.
Slope is change in ychange in x = 0something = 0.
A vertical line is the other way round: the change in x is 0, so its slope is undefined.
Problem 8
What is the equation of the line with slope 2 passing through (0,3)?
- y=3x+2
- y=2x+3
- y=2x-3
- y=x+3
- y=2x
Worked answer
y=2x+3
A line with slope m crossing the y-axis at b has equation y = mx+b.
Here m=2, and the point (0,3) is on the y-axis so b=3.
The equation is y = 2x+3. Swapping the two numbers gives y=3x+2, a different line.
Problem 9
What is the midpoint of the segment joining (2,4) and (8,10)?
- (6,6)
- (5,7)
- (10,14)
- (3,3)
- (4,5)
Worked answer
(5,7)
The midpoint averages the two x values and the two y values separately.
The x value is 2+82 = 5 and the y value is 4+102 = 7.
So the midpoint is (5,7). Adding without halving gives (10,14).
Problem 10
What is the distance from the origin to the point (3,4)?
- 3
- 4
- 5
- 7
- 25
Worked answer
5
The horizontal and vertical distances are 3 and 4, meeting at a right angle at the origin.
By Pythagoras the straight-line distance is √32+42 = √25 = 5.
Adding the coordinates gives 7, which is the distance travelled along the axes rather than directly.