Algebra, Year 10: quadratic, cubic and exponential graphs

The roots of the quadratic function

HintThe same word names the part of a plant that sits at ground level and below.

The whyAt these places y = 0, so they are the solutions of the equation "quadratic = 0". The graph of y = x² − 4 crosses the x-axis at x = −2 and x = 2, and those are the solutions of x² − 4 = 0.

(0, −4)

HintLook for the lowest point of the U: which x-value makes x² as small as it can be?

The whyx² is never negative and is smallest, 0, when x = 0. There y = 0 − 4 = −4, so the curve turns at (0, −4). The curve is symmetrical about the y-axis, and the turning point always lies on the line of symmetry.

3

HintThe two roots are mirror images of each other in the line of symmetry.

The whyThe root x = 1 is 1 unit to the left of the line x = 2, so the other root is 1 unit to the right, at x = 3. Check: 3² − 4 × 3 + 3 = 0. A quadratic graph is symmetrical, so its roots are always equally spaced either side of the turning point.

Draw the line y = 5 and read off the x-values where it meets the curve

HintThe left-hand side of the equation is already on the grid. Ask what graph the right-hand side would make.

The whyEvery point on the curve has a height equal to x² − 2x, so the equation asks where that height is 5. The horizontal line at that height meets the curve twice, at about x = −1.4 and x = 3.4. Readings from a graph are approximate.

maximum

HintDecide whether the turning point is the top of a hill or the bottom of a valley.

The whyFor y = 4 − x², the largest y can be is 4, when x = 0; every other x-value takes something away. So the curve peaks at (0, 4). When the x² term is positive the curve is a U and the turning point is the lowest point instead.

None, because the curve never reaches the x-axis

HintPicture how high the lowest point sits above the horizontal.

The whyThe lowest point of the curve is 3 units above the x-axis, and everywhere else the curve is higher still, so y is never 0. A quadratic graph can cross the x-axis twice, touch it once, or miss it altogether.

A cubic function

HintIts name comes from the solid shape whose volume is found by raising a side length to this power.

The whyy = x³, y = x³ + 2 and y = x³ − 4x are all examples. Their graphs are not U-shaped like a quadratic: they run from one bottom corner of the grid to the opposite top corner, often with an S-shaped wiggle in the middle.

It rises from bottom left to top right, levelling off for a moment as it passes through the origin

HintWork out y for x = −2, −1, 0, 1 and 2 and picture the points.

The whyThe points (−2, −8), (−1, −1), (0, 0), (1, 1) and (2, 8) lie on it. Negative x-values give negative y-values, because a negative number cubed is negative, so the curve has no line of symmetry like a parabola: the left half goes down, not up.

Because 1 ÷ 0 has no value

HintTry to work out the height of the graph at that x-coordinate.

The whyDivision by zero is undefined, so there is nothing to plot on the y-axis. Close to it the values are huge: x = 0.01 gives y = 100, and x = −0.01 gives y = −100. That is why the graph is in two separate pieces.

The top-right and the bottom-left quadrants

HintDecide what sign y has when x is positive, and what sign it has when x is negative.

The whyWhen x is positive, 1/x is positive, so that curve is where both coordinates are positive. When x is negative, 1/x is negative, so the other curve is where both are negative. For example (2, 0.5) and (−2, −0.5) are both on the graph.

3

HintPut the speed into the rule in place of v.

The whyt = 120 ÷ 40 = 3. Doubling the speed to 80 km/h halves the time to 1.5 hours, which is why the curve falls steeply at first and then flattens: each extra km/h saves less time than the one before.

It is the other way up: it falls from top left to bottom right

HintCompare the two y-values at x = 2, and again at x = −2.

The whyEvery y-value changes sign: at x = 2, y = −8 instead of 8, and at x = −2, y = 8 instead of −8. The curve is y = x³ reflected in the x-axis, and it still passes through the origin.

An exponential function

HintThe name comes from another word for an index or power.

The whyy = 2ˣ and y = (1/2)ˣ are examples. Compare y = x², where the variable is the base and the power is fixed: here it is the other way round, and the graph grows (or shrinks) by the same multiplier for every step of 1 in x.

(0, 1), because any positive number to the power 0 is 1

HintPut x = 0 into the rule.

The whyOn the y-axis x = 0, and k⁰ = 1 whatever positive number k is. So y = 2ˣ, y = 10ˣ and y = (1/2)ˣ all cross the y-axis at a height of 1.

To the right it climbs more and more steeply; to the left it gets closer and closer to the x-axis without touching it

HintWork out y for x = 3, 2, 1, 0, −1, −2 and −3 and look at how the values change.

The whyThe values are 8, 4, 2, 1, 1/2, 1/4, 1/8. Going right, y doubles with every step, so the curve gets steeper. Going left, y halves with every step: it shrinks towards zero but a power of 2 is never zero or negative.

decay

HintThe same word describes what happens to a tooth or to a radioactive substance.

The whyFor y = (1/2)ˣ the values at x = 0, 1, 2, 3 are 1, 1/2, 1/4, 1/8: each step multiplies by a number less than 1, so the curve drops quickly at first and then levels off just above the x-axis. When k is greater than 1 the graph rises instead, which is exponential growth.

3

HintSubstitute the coordinates of the point into the rule.

The whyPutting x = 2 and y = 9 gives k² = 9, and the positive number whose square is 9 is 3. So the graph is y = 3ˣ. Check with another point: at x = 1 it should be at height 3.

200; 400; 800

HintEach hour multiplies the number that was there an hour earlier.

The whyDoubling gives 100 × 2 after one hour, then that result × 2 again, and again. The increases are 100, 200 and 400, getting bigger each hour, so the graph is a curve that steepens, not a straight line. The rule is N = 100 × 2ᵗ.