Math Quadratic functions Calculating the intersection

Calculating the intersection

At an intersection, two function graphs intersect.

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Note

The graphs of two quadratic functions can have two, one or no intersection.
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Method

1. Equate both function equations $X_{ S } \Leftrightarrow f(x)=g(x)$
2. Change equation to zero
3. Bring the equation into the reduced quadratic equation
5. Use $x$ in one of the two equations to calculate $y$

Example

Calculate the intersection / intersections of the graphs of $f(x)=-(x+2)^2+1$ and $g(x)=(x+2)^2-1$

1. Equate function equations

$f(x)=g(x)$
$-(x+2)^2+1=(x+2)^2-1$

2. Change equation to zero

$-(x^2+4x+4)+1= x^2+4x+4-1$
$-x^2-4x-4+1= x^2+4x+4-1$
$-x^2-4x-3=x^2+4x+3\quad|+x^2$
$-4x-3= 2x^2+4x+3\quad|+4x$
$-3= 2x^2+8x+3\quad|+3$
$0=2x^2+8x+6$
3. Bring the equation into the reduced quadratic equation

$0=2x^2+8x+6\quad|:2$
$0=x^2+\color{green}{4}x+\color{blue}{3}$

$x_{1,2} = -\frac{\color{green}{p}}{2} \pm\sqrt{(\frac{\color{green}{p}}{2})^2-\color{blue}{q}}$
$x_{1,2} = -2\pm\sqrt{4-3}$
$x_{1,2} = -2\pm\sqrt{1}$
$x_{1,2} = -2\pm1$

$x_{1} = -2+1=-1$
$x_{2} = -2-1=-3$

5. Insert $x_{1}$ and $x_{2}$ in one of the two equations

$g(-1)=(-1+2)^2-1=0$
$g(-3)=(-3+2)^2-1=0$

6. Two intersections:

$S_{1}(-1|0)$ and $S_{2}(-3|0)$