Maths — Pure 1

What does the curve $x^4-y^2=y+1$ look like?

Review question1.1.5

What does the curve $x^4-y^2=y+1$ look like?

Question

Which of the following sketches is a graph of $x^4-y^2=2y+1$ ?

Solution

The equation rearranges to $x^4 = y^2 + 2y + 1 = (y+1)^2$ which is true if and only if $x^2=y+1 \qquad \text{or} \qquad -x^2=y+1.$

So the graph consists of two parabolae, $y=x^2-1$ and $y=-x^2-1$ , which both cross the $y$ -axis at $y=-1$ , and so form the graph in (b).

Curve consisting of two components, one an upward facing parabola with positive y-intercept and the other a downward facing parabola with negative y-intercept

What does the curve

x^4-y^2=y+1

look like?

Question

Which of the following sketches is a graph of

x^4-y^2=2y+1

Solution

The equation rearranges to

x^4 = y^2 + 2y + 1 = (y+1)^2

which is true if and only if

x^2=y+1 \qquad \text{or} \qquad -x^2=y+1.

So the graph consists of two parabolae,

y=x^2-1

and

y=-x^2-1

, which both cross the

y

-axis at

y=-1

, and so form the graph in (b).

Curve consisting of two components, one an upward facing parabola with positive y-intercept and the other a downward facing parabola with negative y-intercept

What does the curve x4−y2=y+1x^4-y^2=y+1x4−y2=y+1 look like?

Question

Solution

What does the curve x4−y2=y+1x^4-y^2=y+1x4−y2=y+1 look like?

Question

Solution

What does the curve $x^4-y^2=y+1$ look like?

What does the curve $x^4-y^2=y+1$ look like?