Intersection of parabolas with the axes

Quadratic equations: Drawing parabolas

Intersection of parabolas with the axes

Intersection with the x-axis

The intersection points of a parabola with the $x$ -axis are the points along the graph where $y=0$ .

To find the $x$ coordinates of these points, equate the formula of the parabola to $0$ and solve for $x$ :

$ax^2+bx+c=0$

This can be done by factorization, completing the square, or using the quadratic formula.

A parabola can have two, one, or no intersections with the $x$ -axis.

geogebra picture

Consider the parabola $y=2x^2+8x-10$ .

The $x$ coordinates of the intersections with the $x$ -axis can be found by solving the equation: $2x^2+8x-10=0$

This is done as follows:

$\begin{array}{rcl}2x^2+8x-10&=&0 \\ &&\phantom{xx}\blue{\text{the equation to be solved}}\\ x^2+4x-5&=&0 \\ &&\phantom{xx}\blue{\text{divided both sides by }2}\\ \left(x+5\right) \left(x-1\right)&=&0 \\ &&\phantom{xx}\blue{\text{factorized}}\\x+5=0 &\lor& x-1=0 \\ &&\phantom{xx}\blue{A \cdot B \Leftrightarrow A=0 \lor B=0}\\x=-5 &\lor& x=1 \\ &&\phantom{xx}\blue{\text{transferred the constants to the right-hand side}} \end{array}$

So the intersections of $y=2x^2+8x-10$ with the $x$ -axis are $\rv{-5,0}$ and $\rv{1,0}$ .

Intersection with the y-axis

The intersection point of a parabola with the $y$ -axis is the point along the graph where $x=0$ .

A parabola always has a single point of intersection with the $y$ -axis.

To find the $y$ coordinate of this point, substitute $0$ for $x$ in the formula of the parabola.

A parabola described by $y=\blue a x^2+\green b x +\purple c$ intersects the $y$ -axis at the point $\rv{0,\purple c}$ .

geogebra picture

Consider the parabola $y=2x^2+8x-10$ .

We find the $y$ coordinate of the intersection with the $y$ -axis by substituting $0$ for $x$ in the formula.

$y=2 \cdot 0^2+8 \cdot 0 -10=-10$

So the the parabola $y=2x^2+8x-10$ intersects the $y$ -axis at the point $\rv{0,-10}$

To determine the graph with the function
$y={{x^2}\over{3}}-7\cdot x$
first draw the intersection point with the $y$ -axis.

The intersection point with the $y$ -axis is equal to the value of the constant in the quadratic function: $0$ . Hence, the coordinates of the intersection point with the $y$ -axis are: $\rv{0,0}$ .

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