Calculus IApplications

Curve sketching, one clue at a time.

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Address: Burnaby, BC, CA
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Build a graph from its domain, intercepts, asymptotes and derivatives. Then practise the full checklist with your own answers checked at every stage.

Build one sketch · Step 1 of 9

Start with the inputs you are allowed to use

f(x)} = {\displaystyle \frac{2x^2}{x^2-1}

Keep x = −1 and x = 1 out of the domain.

Build one sketch · Step 1 of 9

Start with the inputs you are allowed to use

The domain is the set of allowed x-values. Look for division by zero, even roots of negative numbers, and logarithms of zero or negative numbers. We will build one graph through the whole checklist.

Try it: For

{f(x)} = {\frac{2x^2}{x^2-1}}

, which inputs make the denominator zero?

x^2} - {\displaystyle 1} = {\displaystyle (x-1)(x+1)

Exclude −1 and 1. Every other real input works.

(-\infty,-1)\cup(-1,1)

\cup(1,\infty)

The full checklist is domain, intercepts, symmetry, asymptotes, first derivative, second derivative, combine, sketch.

Check yourself

Can an excluded input be a critical number?

Question 1 of 7From class0 of 7 solved

A rational curve with three branches

Sketch

{f(x)} = {\frac{2x^2}{x^2-1}}

. Use

{f'(x)} = {\frac{-4x}{(x^2-1)^2}}

and

{f''(x)} = {\frac{12x^2+4}{(x^2-1)^3}}

. For intervals, use open endpoints and U for a union, for example (-inf,-1)U(-1,1)U(1,inf). Type none for an empty set. List numbers with commas; type coordinates as (x,y).

Work it out on paper first, then type your answer. Stuck? Open one step at a time.

(a)

Find the domain.

Shows here as math.

(-\infty, -1) \cup (-1, 1) \cup (1, \infty)

(b)

Find all x-intercepts: give their x-values. Type numbers separated by commas, or none.

Shows here as math.

0

(c)

Find the y-intercept: give its y-value. Type numbers separated by commas, or none.

Shows here as math.

0

(d)

What symmetry does the function have?

A. Even: y-axis symmetry

(e)

Find the vertical asymptotes: give the x-values of the lines. Type numbers separated by commas, or none.

Shows here as math.

-1,\ 1

(f)

Find the horizontal asymptote (an equation).

Shows here as math.

y } = {\displaystyle  2

(g)

Find the critical numbers. Type numbers separated by commas, or none.

Shows here as math.

0

(h)

Where is f increasing?

Shows here as math.

(-\infty, -1) \cup (-1, 0)

(i)

Where is f decreasing?

Shows here as math.

(0, 1) \cup (1, \infty)

(j)

Give the local maximum as (x,y).

Shows here as math.

\left(0,\ 0\right)

(k)

Find any local minima. Give their x-values. Type numbers separated by commas, or none.

Shows here as math.

\text{none}

(l)

Where is f concave up?

Shows here as math.

(-\infty, -1) \cup (1, \infty)

(m)

Where is f concave down?

Shows here as math.

(-1, 1)

(n)

Find all inflection points. Give their x-values. Type numbers separated by commas, or none.

Shows here as math.

\text{none}

(o)

Which description matches your finished sketch?

A. Two outer branches above y = 2; a middle cap touching the origin

Want a clear plan for sketching any curve?

Bring your practice questions. We work through them together until every type feels routine.

Book a session Next topicNext: Optimization

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Curve sketching, one clue at a time.

Start with the inputs you are allowed to use

Start with the inputs you are allowed to use

A rational curve with three branches

Want a clear plan for sketching any curve?

Questions, answered.

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