LemonmathSelf-study sheet

Grade 12 · Lesson 1.4

Studying and graphing a function

Name: Date: Time spent: min

Goals

Carry out the procedure: domain, limits and asymptotes, table of variations, extrema, intercepts, graph.
Analyse and sketch cubic functions, $y=\dfrac{ax+b}{cx+d}$ and $y=\dfrac{ax^2+bx+c}{mx+n}$ .
Read the features of a function from its graph and use the graph to count the solutions of an equation.
Write the tangent line to a graph at a point.

Remember

Procedure: domain → $y'$ , limits, asymptotes, table of variations → intercepts, special points → draw.
A cubic has either two extrema or none; its centre of symmetry is the point where $y''=0$ .
$y=\dfrac{ax+b}{cx+d}$ : always increasing or always decreasing on each interval; its two asymptotes meet at the centre of symmetry.
$y=\dfrac{ax^2+bx+c}{mx+n}$ : divide to get the oblique asymptote and a tidy derivative.
The number of solutions of $f(x)=k$ is the number of intersections of the graph with the line $y=k$ . Tangent at $x_0$ : $y=f'(x_0)(x-x_0)+f(x_0)$ .

Exercises

Exercise 1

Analyse $y=x^3-3x$ and sketch its graph.

Exercise 2

Analyse $y=\dfrac{x-2}{x+1}$ and sketch its graph.

Exercise 3

Analyse $y=\dfrac{x^2+x+1}{x+1}$ : find its extrema and asymptotes.

Exercise 4

Using the graph of $y=x^3-3x$ (exercise 1), find how many solutions each equation has: a) $x^3-3x=1$ ; b) $x^3-3x=-3$ .

Exercise 5

Consider $y=x^3-6x^2+9x-2$ . Is each claim true or false?

a) The graph meets the y-axis at the point $(0, -2)$ .

b) The local maximum point of the graph is $(1, 2)$ .

c) The equation $x^3-6x^2+9x-2=0$ has exactly two solutions.

d) The centre of symmetry of the graph is the point $(2, 0)$ .

Exercise 6

The tangent to the graph of $y=x^3-3x^2+4$ at the point with x-coordinate $x_0=3$ meets the y-axis at a point. What is the y-coordinate of that point?

Exercise 7

The number of bacteria in a culture after $t$ hours is $N(t)=\dfrac{500t+100}{t+1}$ (thousand), $t\ge0$ . Analyse the function on $[0, +\infty)$ and say: how many bacteria are there at the start, does the number increase or decrease, and what does it approach in the long run?

Exercise 8

The graph of $y=\dfrac{ax+b}{x+c}$ has the vertical asymptote $x=1$ , the horizontal asymptote $y=2$ , and passes through the point $(0, -1)$ . Find $a$ , $b$ , $c$ .

Exercise 9 Review

Find the coordinates of the intersection points of the graph of $y=\dfrac{x+1}{x-1}$ and the line $y=x-1$ .

Exercise 10 Challenge

Write the equations of the tangents to the graph of $y=\dfrac{x+1}{x-1}$ that are parallel to the line $y=-2x$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. The centre of symmetry of the graph of

y=x^3-3x^2+2

is the point

I(2, -2)

.B.

I(0, 2)

.C.

I(1, 0)

.D.

I(-1, -2)

Question 2. Which function has a graph with the vertical asymptote

x=2

and the horizontal asymptote

y=-1

y=\dfrac{-x+1}{x-2}

.B.

y=\dfrac{x+1}{x-2}

.C.

y=\dfrac{-x+1}{x+2}

.D.

y=\dfrac{2x+1}{-x+1}

Question 3. A function

y=f(x)

is continuous on

\mathbb{R}

, increasing on

(-\infty, -1)

and

(3, +\infty)

, decreasing on

(-1, 3)

, with

f(-1)=4

f(3)=-2

\lim\limits_{x\to-\infty}f(x)=-\infty

and

\lim\limits_{x\to+\infty}f(x)=+\infty

. How many solutions does

f(x)+2=0

have?

1

.B.

2

.C.

3

.D.

0

Question 4. The gradient of the tangent to the graph of

y=\dfrac{2x-1}{x+1}

at the point with x-coordinate

x_0=0

1

.B.

-1

.C.

-3

.D.

3

Part II. True or false

Question 1. Let

f(x)=-x^3+6x^2-9x+3

a) $f$ has a local minimum at $x=1$ .	T / F
b) The centre of symmetry of the graph is $I(2, 1)$ .	T / F
c) The equation $f(x)=3$ has three distinct solutions.	T / F
d) The tangent to the graph at $I(2, 1)$ is $y=3x-5$ .	T / F

Part III. Short answer

Question 1. Let

A

and

B

be the two turning points of the graph of

y=x^3-3x^2-9x+1

, and

O

the origin. Find the area of triangle

OAB

Question 2. The tangent to the graph of

y=\dfrac{x+2}{x-1}

at the point with x-coordinate

x_0=3

meets the x-axis at a point. Find the x-coordinate of that point, giving your answer to 1 decimal place.

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: the procedure; an odd function has a graph symmetric about the origin.

Steps:

Compute $y'$ , find the extrema and draw the table of variations.
Find the x-intercepts: $x(x^2-3)=0$ .

Exercise 2.

What you need: $y=\dfrac{ax+b}{cx+d}$ has $y'=\dfrac{ad-bc}{(cx+d)^2}$ , vertical asymptote $x=-\dfrac dc$ and horizontal asymptote $y=\dfrac ac$ .

Steps:

Compute $y'$ and its sign.
Find the two asymptotes and the intercepts.

Exercise 3.

What you need: divide to write $y=x+\dfrac{1}{x+1}$ .

Steps:

Compute $y'=1-\dfrac{1}{(x+1)^2}$ and write it as one fraction.
Draw the table of variations with $-2$ , $-1$ , $0$ marked.

Exercise 4.

What you need: the number of solutions of $f(x)=k$ is the number of intersections of the graph with the line $y=k$ .

Steps:

Compare $k$ with the local maximum $2$ and the local minimum $-2$ .
Count the intersections.

Exercise 5.

What you need: the table of variations of a cubic; the centre of symmetry has the root of $y''=0$ as its x-coordinate.

Steps:

Compute $y'$ and find the extrema.
Compare $0$ with the local maximum and minimum values to count solutions.
Solve $y''=0$ .

Exercise 6.

What you need: the tangent at $x_0$ is $y=f'(x_0)(x-x_0)+f(x_0)$ (derivatives, grade 11).

Steps:

Compute $f(3)$ and $f'(3)$ .
Write the tangent and put $x=0$ .

Exercise 7.

What you need: the sign of $N'(t)$ tells whether it increases or decreases; the horizontal asymptote gives the long-run value.

Steps:

Compute $N(0)$ and $N'(t)$ .
Compute $\lim\limits_{t\to+\infty}N(t)$ .

Exercise 8.

What you need: $y=\dfrac{ax+b}{x+c}$ has the vertical asymptote $x=-c$ and the horizontal asymptote $y=a$ .

Steps:

Deduce $c$ and $a$ from the two asymptotes.
Substitute the point $(0, -1)$ to find $b$ .

Exercise 9.

What you need: the x-coordinates of the intersections solve $\dfrac{x+1}{x-1}=x-1$ with $x\neq1$ ; solve a quadratic equation (grade 10).

Steps:

Multiply both sides by $x-1$ and get a quadratic equation.
Check the condition $x\neq1$ , then compute the y-coordinates.

Exercise 10.

What you need: parallel lines have the same gradient; the gradient of the tangent at $x_0$ is $y'(x_0)$ .

Steps:

Solve $y'(x_0)=\dfrac{-2}{(x_0-1)^2}=-2$ .
For each root, compute $y_0$ and write the tangent; check that it is not the line $y=-2x$ itself.

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