Sketching a Curve from Its Factorised Form

Mathematics · Algebra · ages 16-17

Name ______________________ Date ____________

The curve y = (x - 1)(x + 4)² crosses the x-axis at x = -4.
Circle one: True False
y = (x - 2)(x + 1)(x + 6). What happens for large negative x?
- y climbs to large positive values
- y falls to large negative values
- y stays near zero
- y settles at the y-intercept
For y = (x + 1)(x - 2)(x - 4), what is y when x = 5?
Answer: ______________
What is the y-intercept of y = (x + 1)(x - 2)(x - 4)?
Answer: ______________
What is the degree of y = x(x - 1)(x + 2)(x - 3)?
Answer: ______________
Why does y = x(x - 3)² touch the x-axis at x = 3 instead of crossing it?
- The factor (x - 3) is squared, so its multiplicity is even
- The factor x has degree 1, so it pulls the graph through
- The leading coefficient is positive, so the graph bounces
- The constant term is zero, so the graph flattens
y = -(x + 2)(x - 3)(x - 5). What happens for large positive x?
- y climbs to large positive values
- y settles at -30
- y falls to large negative values
- y stays just below the axis
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
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

False · (x + 4) is repeated, so near -4 the squared part stays positive and y keeps its sign. The curve touches there and turns back.
y falls to large negative values · Three brackets give a leading term of x³ with a positive sign. Cubing a large negative number gives a large negative number, so the left end falls away.
18 · y = (6)(3)(1) = 18 at x = 5.
8 · Substitute x = 0: (1)(-2)(-4) = 8.
4 · One factor from each bracket gives degree 1 + 1 + 1 + 1 = 4.
The factor (x - 3) is squared, so its multiplicity is even · A squared factor has even multiplicity 2, and even multiplicity touches rather than crosses.
y falls to large negative values · The brackets give x³, and the minus in front makes the leading term -x³. A large positive x then gives a large negative y.
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.
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.