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Longhand

Before the midterm

Letting an even power change sign with x

The habit: x², x4 and (x − a)² are never negative; only odd exponents flip across zero.

5 practice questions are built to catch it, in 4 topics.

Practise against it: 4 questions

What it looks like

A question from power functions and first sketches, with the answer this slip gives beside the right one.

Capstone5 marks · 9 min

Let y=kx2+x3y = kx^{2} + x^{3}, where kk is a nonzero constant. Which statement about the graph is correct for both signs of kk?

With the slip
The graph crosses the x-axis at the origin, changing sign there, and crosses it again at x = −k. Near the origin it looks like y = kx², and far from the origin it looks like y = x³, falling on the left and rising on the right.
Worked through
The graph touches the x-axis at the origin without crossing it and crosses it once more, at x = −k. Near the origin it looks like the parabola y = kx², opening up if k > 0 and down if k < 0, and far from the origin it looks like y = x³, falling on the left and rising on the right.

x² is never negative, so y = x²(x + k) has the sign of x + k on both sides of 0: the graph meets the axis at the origin and turns back, staying on one side.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    Evaluate the limit. lim⁡x→−3x−1(x+3)2\displaystyle \lim_{x \to -3} \frac{x - 1}{(x + 3)^{2}}

    LimitsIntroductory

  2. 02
    Let f(x,y)=9−x2−y2f(x, y) = \sqrt{9 - x^{2} - y^{2}}. Which statement gives f(2,−1)f(2, -1) and the domain of ff correctly?

    Functions of two variables and partial derivativesIntroductory

  3. 03
    A student describes the graph of the function below for large negative xx as shown. Find the first line that is wrong. y=x2−x6\displaystyle y = x^{2} - x^{6}

    Power functions and first sketchesStandard

  4. 04
    Let f(x)=x−23+1f(x) = \sqrt[3]{x - 2} + 1. Which statement describes the graph of ff at x=2x = 2?

    Curve sketchingStandard

Start the set