Sketching a Curve from Its Factorised Form

Mathematics · Algebra · ages 16-17

Name ______________________ Date ____________

y = x(x - 6)². At which value of x does the curve touch the axis without crossing it?
Answer: ______________
y = (x - 4)(x + 1)(x + 5) has three separate x-axis crossings.
Circle one: True False
y = (x - 1)(x - 2)(x - 3). What happens to y for large positive x?
- it settles at -6
- it falls away to large negative values
- it flattens out at zero
- it climbs to large positive values
What is the y-intercept of y = (x + 3)(x - 1)(x - 2)?
Answer: ______________
For y = (x + 3)(x - 1)(x - 4), is the curve above or below the x-axis between x = 1 and x = 4?
- above, every bracket is positive there
- below, the product comes out negative at x = 2
- above, because two of the roots are positive
- below, because the y-intercept is negative
y = (x - 1)(x - 2)(x - 3)² meets the x-axis four times.
Circle one: True False
y = (x + 1)²(x - 2)². What do the two ends of this curve do?
- the left end climbs and the right end falls
- both ends climb
- both ends fall
- the left end falls and the right end climbs
Jo says y = (x + 2)²(x - 4) crosses the x-axis three times because the top power is 3. Where is the slip?
- the top power is 2, so there are only two roots
- the squared bracket gives no root, so only 4 counts
- a cubic must cross the axis three separate times
- the repeated bracket touches, so the axis is met twice
y = (x - 1)²(x + 5)² meets the x-axis at two places. At how many of them does the curve cross the axis rather than touch it?
Answer: ______________
Ben sketches y = -(x + 1)(x - 2)(x - 3) climbing on the right, because two of the roots are positive. Where is the slip?
- the roots are -1, 2 and 3, so two of them are negative
- the minus sign only flips the y-intercept
- the ends come from the top power and its sign
- the y-intercept is what decides which way the ends go

Answer key

For grown-ups. Fold this page away before handing over the rest.

Sketching a Curve from Its Factorised Form W1-mt_x4tZLdH8fh-s1

6 · The repeated bracket is (x - 6), so the touch is at x = 6. The single factor x gives an ordinary crossing at 0.
True · The three brackets give the roots 4, -1 and -5. They are all different, so there are three separate crossings.
it climbs to large positive values · For large x every bracket is a large positive number, and three positives multiply to a large positive. The leading term is x³.
6 · Put x = 0: (0 + 3)(0 - 1)(0 - 2) = 3 times -1 times -2 = 6.
below, the product comes out negative at x = 2 · At x = 2 the brackets are 5, 1 and -2. One negative among three factors makes the product negative, so the curve dips below the axis in that gap.
False · The brackets give the roots 1, 2, 3 and 3. The last two are the same value, so the curve meets the axis at three places, touching at 3.
both ends climb · Two squared brackets give x⁴, an even power with a positive sign, so both ends go the same way and that way is up.
the repeated bracket touches, so the axis is met twice · The top power is 3, but two of those factors are the same bracket. The curve touches at -2 and crosses at 4, so it meets the axis at two places.
0 · Both brackets are squared, so each stays positive on both sides of its root and y never changes sign. The curve touches at 1 and at -5 and crosses nowhere.
the ends come from the top power and its sign · Where the roots sit says nothing about the ends. Three brackets give x³, the minus makes it -x³, and a large positive x then gives a large negative y.