1. Tangent & Normal Lines

The Tangent touches the curve ($m_{tan} = f'(x)$).

The Normal is perpendicular ($m_{norm} = -1/m_{tan}$).

Tangent Line Normal Line
1.0
1. Function: f(x) = x²
2. Derivative: f'(x) = 2x
3. Substitute x = 1:
Tangent Slope ($m$) = 2(1) = 2
4. Normal Slope ($m_\perp$) = $-1/m$ = -0.5

2. You Pick the Point

Select a curve and a point. The Tangent Line is drawn for you.

Tangent Visualization
1
Step A: Find y
Substitute $x$ into ...
$y =$
Step B: Find Slope ($m$)
Find $f'(x)$ and subs $x$.
$m =$

3. Complete Challenge

Find the equations. You can enter fractions like 1/2 or -3/4.

Given Curve: $$ y = x^2 + 2x $$
At the point where x = 1
Step 1: Coordinates (Find $y_1$)
$y_1 =$
Step 2: Gradient (Find $m$)
$m =$
Step 3: Tangent Equation ($y = mx + c$)
$y =$ $x +$
Step 4: Normal Equation ($y = m_{\perp}x + c$)
$y =$ $x +$