How do you find an equation of the tangent line to the parabola

Question

Determine the equation of the line that is tangent to the parabola with equation

$y = x^2 − 2x + 2$

at the point $(3, 5)$

Answer 1

The directional coefficient equals $\frac{dy}{dx} = 2x-2$ evaluated at $x=3$.

Answer 2

1. Find $\frac{dy}{dx}$.

2. Insert $x=3$ into $\frac{dy}{dx}$. This will give you the gradient which equals $a$ of the tangent equation $ax+c$.

3. Next, insert $y=5$ into your tangent equation and rearrange to find $c$.

Try to do it yourself but ask questions if you get stuck.

Answer 3

For fun:

0) $y-5=m(x-3)$; Line passes through $(3,5)$;

1) Intersection of line with $y=x^2-2x+2$;

$m(x-3)+5=x^2-2x+2;$

$x^2-x(2+m)+(3m-3)=0$;

2) Discriminant:

$D= (2+m)^2-4(3m-3)=0$;

$m^2-8m+16=0;$

$(m-4)^2=0$; $m=4$;

3) Tangent:

$y-5=4(x-3).$