AI SL · 12.4
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Where two graphs meet
12.4.6
Screenshot and video to come
Screenshot and video to come
The curve \(y = 2x^2 − 3x − 1\) and the line \(y = x + 4\) meet at two points. Find the coordinates of both points.
Have a go first. Then open the working one line at a time.
Graph y = 2x² − 3x − 1 and y = x + 4
The curve and the line have the same y-value where they meet.
From the calculator: x = −0.871 and x = 2.87
Check with the solver: 2x² − 4x − 5 = 0 gives the same two values.
y = x + 4 gives y = 3.13 and y = 6.87
Use the line to find y: it is the simpler equation. A point needs both coordinates.
(−0.871, 3.13) and (2.87, 6.87)
Check one point in the curve: the y-value should match.
A company sells \(x\) mugs a week. The weekly cost is \(C = 2400 + 18x\) dollars and the weekly revenue is \(R = 95x − 0.4x^2\) dollars. (a) Find the break-even quantities. (b) What is the smallest whole number of mugs that gives a profit?
Have a go first. Then open the working one line at a time.
Graph C = 2400 + 18x and R = 95x − 0.4x²
Break-even is where cost and revenue are equal.
From the calculator: x = 39.1 and x = 153
Two break-even points.
R > C for 39.1 < x < 153
The revenue curve is above the cost line between the two points.
P(39) = −5.40, P(40) = 40.00; the last profit is at 153
Mugs come in whole numbers. 40 to 153 mugs make a profit.
Graph P = R − C: maximum at x = 96.3; P(96) = 1305.60
Check the whole numbers either side of the maximum.
The population of town A is \(P = 12 000 + 450t\) and the population of town B is \(P = 8000 × 1.06^t\), where \(t\) is the number of years from now, \(t \geq 0\). (a) When are the populations equal? (b) Which town is larger after 30 years?
Have a go first. Then open the working one line at a time.
Graph y = 12 000 + 450x and y = 8000 × 1.06^x
Use x for t on the calculator. The equation has no polynomial form, so the graph does the work.
From the calculator: t = 14.3 population about 18 456
One crossing for t ≥ 0.
A second crossing at t = −21.6 is before now
It is outside t ≥ 0, so reject it. Always check the domain.
At t = 30: A = 25 500 B = 45 948
After the crossing the exponential is above the line: town B is larger.
A ball is thrown upwards from a balcony. Its height above the ground after \(t\) seconds is \(h = −4.9t^2 + 14t + 1.2\) metres. (a) Find the greatest height and when it is reached. (b) When does the ball hit the ground? (c) For how long is the ball higher than 8 m?
Have a go first. Then open the working one line at a time.
Graph y = −4.9x² + 14x + 1.2 for x ≥ 0
x is t and y is h.
From the calculator: maximum at t = 1.43, h = 11.2
The top of the curve.
h = 0: t = 2.94 (the other root is negative)
The ground is the x-axis. Reject the negative time.
Intersections with y = 8: t = 0.620 and t = 2.24
Above 8 m between these two times.
2.24 − 0.620 = 1.62 s
Use the unrounded values from the calculator.
A cup of tea cools. Its temperature after \(t\) minutes is \(T = 21 + 64 × 0.92^t\) degrees Celsius, \(t \geq 0\). (a) Find the starting temperature. (b) What temperature does the tea approach? (c) When does the tea reach 40 °C?
Have a go first. Then open the working one line at a time.
Graph y = 21 + 64 × 0.92^x
A decaying curve that flattens out.
At t = 0: T = 21 + 64 = 85 °C
The value at the start is the y-intercept.
T = 21
0.92^t gets closer to 0, so T gets closer to 21 and never reaches it. 21 °C is room temperature.
Intersection with y = 40: t = 14.6 minutes
Graph the line y = 40 and find where the curve meets it.