The equation of the tangent

Algebra and calculus, grade 10 (4 units) Lesson 12 of 20Learning

The equation of the tangent

The tangent’s equation in three steps

The tangent to a graph at a point is a line: its equation needs one point on it and its slope. The derivative gives the slope, and the function gives the point.

$\displaystyle x_1$ – the $\displaystyle x$ -coordinate of the point of contact (where the tangent touches the graph).
$\displaystyle {y_1}={f(x_1)}$ – its $\displaystyle y$ -coordinate: substitute in the function, not in the derivative.
$\displaystyle {m}={f'(x_1)}$ – the slope of the tangent: substitute in the derivative.

\displaystyle y-y_1

\displaystyle =

\displaystyle m(x-x_1)

Example: the tangent at a given point

The example below finds the equation of the tangent to the graph of $\displaystyle {f(x)}={x^2-4x+5}$ at the point where $\displaystyle {x}={3}$ .

\displaystyle f(x)=x^{2}-4x+5

\displaystyle y=2x-4

\displaystyle A=(3,2)

The point: $\displaystyle {f(3)}={9-12+5}={2}$ , so $\displaystyle (3,2)$ .
The derivative: $\displaystyle {f'(x)}={2x-4}$ , and the slope: $\displaystyle {f'(3)}={2}$ .
The equation: $\displaystyle {y-2}={2(x-3)}$ , so $\displaystyle {y}={2x-4}$ – the dashed line.

The point where the tangent has a given slope

When the slope is given and the point is not, solve the equation $\displaystyle {f'(x)}={m}$ : its solutions are the $\displaystyle x$ -coordinates of the points where the tangent has that slope.

The example below finds the tangents to the graph of $\displaystyle {f(x)}={x^3-3x}$ whose slope is 9:

f'(x)=3x^2-3=9\quad\Rightarrow\quad x^2=4

So $\displaystyle {x}={2}$ or $\displaystyle {x}={-2}$ .

At $\displaystyle {x}={2}$ : $\displaystyle {f(2)}={2}$ , and the tangent is $\displaystyle {y}={9x-16}$ .
At $\displaystyle {x}={-2}$ : $\displaystyle {f(-2)}={-2}$ , and the tangent is $\displaystyle {y}={9x+16}$ .
Both tangents have the same slope, so they are parallel.

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

\displaystyle y=9x-16

\displaystyle y=9x+16

\displaystyle A=(2,2)

\displaystyle B=(-2,-2)

A horizontal tangent and a tangent parallel to a line

Two cases of a given slope:

A horizontal tangent (parallel to the $\displaystyle x$ -axis) – its slope is 0: solve $\displaystyle {f'(x)}={0}$ , and the tangent is $\displaystyle {y}={f(x_1)}$ .
A tangent parallel to the line $\displaystyle {y}={mx+b}$ – parallel lines have the same slope: solve $\displaystyle {f'(x)}={m}$ .
Where a tangent meets the axes: put $\displaystyle {x}={0}$ (the $\displaystyle y$ -axis) or $\displaystyle {y}={0}$ (the $\displaystyle x$ -axis) in the tangent’s equation.

Exercise 1

The function is $\displaystyle {f(x)}={x^2+2x}$ .
The point $\displaystyle (1,3)$ is on its graph.

Work out the slope of the tangent to the graph at the point $\displaystyle (1,3)$ .

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

/ – 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

Write the equation of the tangent at this point on the graph board.

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

\displaystyle (1,3)

Exercise 2

The function is $\displaystyle {f(x)}={x^3-2x^2}$ .
Write the equation of the tangent to the graph at the point where $\displaystyle {x}={2}$ on the graph board.

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

\displaystyle (2,0)

Exercise 3

The function is $\displaystyle {f(x)}={x^2-6x+7}$ .
The tangent to the graph at the point $\displaystyle A$ is parallel to the line $\displaystyle {y}={2x+3}$ (the dashed one).

Write an equation and solve it: what is the $\displaystyle x$ -coordinate of the point $\displaystyle A$ ?

Write the equation of the tangent at the point $\displaystyle A$ on the graph board.

\displaystyle f(x)=x^{2}-6x+7

\displaystyle y=2x+3

Exercise 4

The function is $\displaystyle {f(x)}={x^3-12x}$ .

At the points where the tangent to the graph is horizontal, $\displaystyle {f'(x)}={0}$ .
Solve the equation.

Write the equations of the two horizontal tangents on the graph board.

\displaystyle f(x)=x^{3}-12x

Exercise 5

The function is $\displaystyle {f(x)}={(2x-3)^2}$ .

Write the equation of the tangent to the graph at the point $\displaystyle (1,1)$ on the graph board.

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

\displaystyle (1,1)

Solve an equation: at which point does the tangent meet the $\displaystyle x$ -axis?

The tangent meets the $\displaystyle y$ -axis at the point $\displaystyle (0,5)$ .
Work out the area of the triangle the tangent makes with the axes.

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Formulas: algebra and calculus, grade 10 (4 units)

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Algebraic technique

The square of a sum
$\displaystyle (a+b)^2$ $\displaystyle =$ $\displaystyle a^2+2ab+b^2$
The square of a difference
$\displaystyle (a-b)^2$ $\displaystyle =$ $\displaystyle a^2-2ab+b^2$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
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 roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

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

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
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 derivative of a product
$\displaystyle (f(x)\cdot g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

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
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

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
$\displaystyle f'(x)=0$