# Parabola Calculator

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Parabolas are the graphs of quadratic functions. All parabolas are “U” shaped and have a highest or lowest point that is called the vertex. Vertex is the highest point if the parabola is opened down and is the lowest point if the parabola is opened up. Parabolas always have a single $y$ - intercept and may or may not have $x$ - intercept. Every parabola has an axis of symmetry which divided the graph in two parts which are mirror image of each other.

The standard form of a Parabola equation is a quadratic equation $f(x)= ax^{2} + bx + c$

The vertex form of the Parabola equation is given by

$y = ax - h^{2} + k$

Where $(h, k)$ is the vertex of the parabola,

$(x, y)$ is a point on parabola,

and ‘a’ is the stretch/compression of the equation.
• Parabolas open upwards or downwards.
• The “zero” or “x – intercept” or “root” of a parabola is where the graph crosses the x – axis.
• The point where the graph crosses the y – axis is y-intercept of a parabola.
• The axis of symmetry divides the parabola into two equal halves mirror image to each other.
• The point where the axis of symmetry and the parabola meet is called the vertex of a parabola and It is at this point where the parabola obtains its maximum or minimum value.
• The value of the y co-ordinate of the vertex is called the optimal value.

## How To Solve Parabola Equations

Below you could see the steps

Step 1: Identify the form in which the given Parabolic equation is.
Step 2: If the equation is given in standard form use any of the three methods mentioned below to solve for the parabola.
• By factorization
• By completing the squares
Step 3: If the equation is given in Vertex form then find the x-intercept by plugging $y = 0$ to solve for parabola.
Step 4: Find the vertex of the parabola.
• If the given equation is in the vertex form $y = ax - h^{2} + k$  then $(h,k)$ is the vertex of the parabola.
• If the given equation is in standard form  $f(x)= ax^{2} + bx + c$ then $h =$ $\frac{-b}{2a}$ and $k = f h$.
Step 5: If the value of $a > 0$, the parabola opens up and if $a < 0$, then the parabola opens downwards.
Step 6:
Find the axis of symmetry which is given by $x = h$ or $x =$ $\frac{-b}{2a}$.

## Parabola Examples

Below you could see some examples

### Solved Examples

Question 1: What is the vertex of a parabola whose equation is given by $y = 3x - 2^{2} + 5$ ? Does the parabola open up or down?
Solution:
The given equation of a parabola is $y = 3x - 2^{2} + 5$. This is in the vertex form.

Comparing it to the vertex form $y = ax - h^{2} + k$, we get $h = 2$ and $k = 5$.

Thus the vertex $(h,k)$ = $(2,5)$. Also $a = 3$ which is positive.

If $a >0$ then the parabola opens up. So the give parabola opens upwards.

Question 2: Convert the given parabola equation $y = 2x^{2} - 12x + 20$ to a vertex form.
Solution:
The given equation is $y = 2x^{2} - 12x + 20$.

To convert it into vertex form we use the method of completion of squares.

$y = 2x^{2} - 12x + 20$
$y = 2(x^{2} - 6x) + 20$
$y = 2(x^{2} - 2 \times x\times 3) + 20$
$y = 2x^{2} - 2 \times x \times 3 + 3^{2} - 2 \times 3^{2} + 20$
$y = 2x - 3^{2} - 18 + 20$
$y = 2x - 3^{2} + 2$

$y = 2x - 3^{2} + 2$ which is the vertex form of the given parabola.

Question 3: Solve the parabola whose equation is given $y = x^{2} + 2x - 15$. Also find the vertex and the axis of symmetry.
Solution:
The given parabola is in quadratic form. To solve for its zeros we plug $y = 0$ and solve for $x$.

Thus $x^{2} + 2x - 15 = 0$. We use factorization method to solve for $x$.

$x^{2} + 5x - 3x - 15 = 0$
$(x + 5) (x - 3) = 0$

By zero product rule, $x + 5 = 0$ or $x - 3 = 0$

Thus x = -5 or 3 or the zeros are at $(0, -5) (0, 3)$.

If the given equation is in standard form $f(x)= ax^{2} + bx + c$ then $h$ = $\frac{-b}{2a}$ and $k = f h$

Here a = 1, b = 2 and c = -15

So, $h$ = $\frac{-b}{2a}$ = $\frac{-2}{2}$ = $-1$

And, $k = f h = f(-1) = -1^{2} + 2(-1) - 15 = 1 - 2 - 15 = 16$

Thus the vertex = $(-1, -16)$

The axis of symmetry is given by $x = h \rightarrow x = -1$ or $x + 1 = 0$