The graph of a function: zeros, sign, rising and falling

Calculus, grade 10 (5 units) Lesson 1 of 26Learning

The graph of a function: zeros, sign, rising and falling

A function and its graph

A function gives every $\displaystyle x$ in its domain one value of $\displaystyle y$ . Its graph is all the points $\displaystyle (x,f(x))$ .

So every vertical line meets the graph of a function at one point at most.
The line $\displaystyle {x}={3}$ is not the graph of a function: every value of $\displaystyle y$ goes with $\displaystyle {x}={3}$ on it.
The circle $\displaystyle {x^2+y^2}={25}$ is not the graph of a function either: both $\displaystyle {y}={4}$ and $\displaystyle {y}={-4}$ go with $\displaystyle {x}={3}$ on it.

What a graph tells

The graph of a function tells five things:

The points where it meets the axes: the $\displaystyle x$ -axis – the points where $\displaystyle {f(x)}={0}$ (the zeros); the $\displaystyle y$ -axis – the point $\displaystyle (0,f(0))$ .
Where it is positive: where the graph is above the $\displaystyle x$ -axis, $\displaystyle f(x)>0$ . Where it is negative: where it is below it, $\displaystyle f(x)<0$ .
Where it rises: where $\displaystyle y$ grows as $\displaystyle x$ grows. Where it falls: where $\displaystyle y$ gets smaller.
A maximum point: a point where the function goes from rising to falling. A minimum point: from falling to rising. Both are extremum points.

On the graph below, of $\displaystyle {f(x)}={x(x-3)^2}$ :

\displaystyle f(x)=x(x-3)^{2}

\displaystyle A=(1,4)

\displaystyle B=(3,0)

The points where it meets the axes: $\displaystyle (0,0)$ and $\displaystyle (3,0)$ .
Positive where $\displaystyle 0<x<3$ and where $\displaystyle x>3$ ; negative where $\displaystyle x<0$ .
Rises where $\displaystyle x<1$ and where $\displaystyle x>3$ ; falls where $\displaystyle 1<x<3$ .
$\displaystyle (1,4)$ – a maximum point; $\displaystyle (3,0)$ – a minimum point.

At $\displaystyle {x}={3}$ the graph touches the $\displaystyle x$ -axis without crossing it, so the function is positive on both sides of 3.

The linear function

The graph of $\displaystyle {f(x)}={mx+b}$ is a straight line: $\displaystyle m$ is its slope, and $\displaystyle b$ where it meets the $\displaystyle y$ -axis.

$\displaystyle m>0$ : the function rises everywhere. $\displaystyle m<0$ : it falls everywhere. $\displaystyle {m}={0}$ : it is constant.
Its zero is the solution of $\displaystyle {mx+b}={0}$ ; on one side of it the function is positive, on the other negative.
A linear function with $\displaystyle m\neq0$ has no extremum points.

The quadratic function

The graph of $\displaystyle {f(x)}={ax^2+bx+c}$ is a parabola. Everything the graph tells can be worked out:

The zeros: solve $\displaystyle {ax^2+bx+c}={0}$ by factoring or by the quadratic formula.
The vertex is the only extremum point, and its $\displaystyle x$ -coordinate is $\displaystyle {x}={\frac{-b}{2a}}$ .
$\displaystyle a>0$ : the parabola opens upwards, the vertex is a minimum, and the function is negative only between its zeros.
$\displaystyle a<0$ : the parabola opens downwards, the vertex is a maximum, and the function is positive only between its zeros.

The example below reads $\displaystyle {f(x)}={x^2-2x-8}$ this way:

$\displaystyle {x^2-2x-8}={(x-4)(x+2)}$ , so the zeros are $\displaystyle {x}={-2}$ and $\displaystyle {x}={4}$ .
$\displaystyle {x}={\frac{2}{2}}={1}$ and $\displaystyle {f(1)}={1-2-8}={-9}$ : the vertex $\displaystyle (1,-9)$ is a minimum point.

\displaystyle f(x)=x^{2}-2x-8

\displaystyle V=(1,-9)

$\displaystyle {a}={1>0}$ , so the function is negative where $\displaystyle -2<x<4$ , and positive where $\displaystyle x<-2$ and where $\displaystyle x>4$ .
It falls where $\displaystyle x<1$ , and rises where $\displaystyle x>1$ .

Exercise 1

The board below shows the graph of $\displaystyle {f(x)}={-(x+1)^2(x-2)}$ .

Mark on the graph the points where it meets the $\displaystyle x$ -axis, and the extremum points.

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

Write where $\displaystyle f$ is positive and where it is negative.
Write where $\displaystyle f$ rises and where it falls.

Exercise 2

The function is $\displaystyle {f(x)}={-x^2+6x-5}$ .

Find the zeros of $\displaystyle f$ .

The $\displaystyle x$ -coordinate of the vertex is $\displaystyle {x}={\frac{-6}{-2}}={3}$ .
Work out its $\displaystyle y$ -coordinate.

$\displaystyle f(x)=-x^{2}+6x-5$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the inequality $\displaystyle f(x)>0$ : where is $\displaystyle f$ positive?

Mark the zeros and the vertex on the graph, and check your answers.

\displaystyle f(x)=-x^{2}+6x-5

Exercise 3

The function is $\displaystyle {f(x)}={-2x+6}$ .

Find the zero of $\displaystyle f$ .

Solve the inequality $\displaystyle f(x)<0$ : where is $\displaystyle f$ negative?

Does $\displaystyle f$ rise or fall?
Explain by its slope.

Exercise 4

The functions are $\displaystyle {f(x)}={x^2-1}$ and $\displaystyle {g(x)}={x+1}$ .

Mark on the board the points where the two graphs meet.

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

\displaystyle g(x)=x+1

Work out the $\displaystyle x$ -coordinates of the meeting points.

Solve the inequality $\displaystyle f(x)<g(x)$ .
Check on the board: where is the parabola below the line?

Exercise 5

Which of these lines is the graph of a function?
Explain each answer.

$\displaystyle {y}={x^2-4}$
$\displaystyle {x}={-2}$
$\displaystyle {x^2+y^2}={9}$
$\displaystyle {y}={5}$

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$

The properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units right ( $\displaystyle p<0$ – left)
$\displaystyle y=f(x-p)$
A vertical stretch ( $\displaystyle |a|>1$ ) or squeeze ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$
A horizontal squeeze by $\displaystyle a$ ( $\displaystyle a>1$ ) or stretch ( $\displaystyle 0<a<1$ )
$\displaystyle y=f(ax)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$