How to find quadratic equation from graph without vertex

How to find quadratic equation from graph without vertex

To find the vertex of a parabola, you first need to know how to graph quadratic equations. When graphing these, remember that every quadratic function can be put into a standard form (more on this later). This allows you to find the leading coefficient and solve for the x-intercepts. The x-intercept and y-intercept are points on the graph where the parabola intersects the x-axis or y-axis.

Putting the quadratic function into standard form will also let you find the axis of symmetry, the line that runs through the vertex and divides the parabola in half. You can then find the x-coordinate and y-coordinate of the vertex, which is the highest or lowest point on a parabola.

Definition of a Parabola

A parabola is a set of points that are equal distances from both a focus (a fixed point) and a directrix (a fixed line). It’s the “u” shape that forms when one graphs a quadratic equation or quadratic function.

Depending on the coefficients of the original equation, the parabola opens to the right side, to the left side, upwards, or downwards.

The Axis of Symmetry of a Parabola

Before we find the vertex of a parabola, let’s review the axis of symmetry.

Remember, in a parabola, every point represents an x and a y that solves the quadratic function.

The axis of symmetry is the vertical line that goes through the vertex of a parabola. The vertex of the parabola is the maximum or minimum point on the graph of the quadratic function.

Remember that every quadratic function can be written in the standard form

How to find quadratic equation from graph without vertex
.

The equation for the axis of symmetry of a parabola can be expressed as:

How to find quadratic equation from graph without vertex

Finding the Vertex of the Parabola

To find the coordinates for the vertex of the parabola, you should first use the equation to find the axis of symmetry. Then, substitute the x value that you find back into the original question to get the y-value.

Let’s solve for the axis of symmetry when a = 1 and b = —4.

How to find quadratic equation from graph without vertex

Now, we know that x = 2. Now we substitute that back into the original quadratic equation.

How to find quadratic equation from graph without vertex

How to find quadratic equation from graph without vertex

How to find quadratic equation from graph without vertex

Solving it gives us y = -1. We now know that the vertex of the parabola is the coordinate (2, -1). Finding the vertex of a parabola couldn’t be easier once you know these steps!

Find the Vertex of a Parabola in No Time

To find the vertex of a parabola, you first need to find x (or y, if your parabola is sideways) through the formula for the axis of symmetry. Then, you’ll use that value to solve for y (or x if your parabola opens to the side) by using the quadratic equation. Those two coordinates are your parabola’s vertex.

More Math Homework Help

  • How To Use the Leading Coefficient Test To Graph End Behavior
  • One-to-One Functions: The Exceptional Geometry Rule
  • How To Master Quadratic Regression
  • How To Find the Axis of Symmetry of a Parabola

Parabolas


The graph of a quadratic equation in two variables (y = ax2 + bx + c) is called a parabola. The following graphs are two typical parabolas their x-intercepts are marked by red dots, their y-intercepts are marked by a pink dot, and the vertex of each parabola is marked by a green dot:

How to find quadratic equation from graph without vertex
How to find quadratic equation from graph without vertex

We say that the first parabola opens upwards (is a U shape) and the second parabola opens downwards (is an upside down U shape). In order to graph a parabola we need to find its intercepts, vertex, and which way it opens.

Given y = ax2 + bx + c , we have to go through the following steps to find the points and shape of any parabola:

  1. Label a, b, and c
  2. .
  3. Decide the direction of the paraola
  4. :

    If a > 0 (positive) then the parabola opens upward.
    If a < 0 (negative) then the parabola opens downward.

  5. Find the x-intercepts
  6. :


    Notice that the x-intercepts of any graph are points on the x-axis and therefore have y-coordinate 0. We can find these points by plugging 0 in for y and solving the resulting quadratic equation (0 = ax2 + bx + c). If the equation factors we can find the points easily, but we may have to use the quadratic formula in some cases. If the solutions are imaginary, that means that the parabola has no x-intercepts (is strictly above or below the x-axis and never crosses it). If the solutions are real, but irrational (radicals) then we need to approximate their values and plot them.

  7. Find the y-intercept
  8. :

    The y-intercept of any graph is a point on the y-axis and therefore has x-coordinate 0. We can use this fact to find the y-intercepts by simply plugging 0 for x in the original equation and simplifying. Notice that if we plug in 0 for x we get: y = a(0)2 + b(0) + c or y = c. So the y-intercept of any parabola is always at (0,c).

  9. Find the vertex (h,k)
  10. :


    To find the x-coordinate for the vertex we use the following formula:

    How to find quadratic equation from graph without vertex

    To find the y-coordinate for the vertex we plug in h in the original equation:
    k = a(h)2 + b(h) + c

  11. Plot the points and graph the parabola 

Example 1)

Graph y = x2 + 2x - 8  

In this problem: a = 1, b = 2 , and c = -8.

Since "a" is positive we'll have a parabola that opens upward (is U shaped).
To find the x-intercepts we plug in 0 for y:
0 = x2 + 2x - 8 (which factors)
0 = (x + 4)(x - 2)
x = -4 or x = 2
So this parabola has two x-intercepts: (-4,0) and (2,0).

To find the y-intercept we plug in 0 for x:
y = (0)2 + 2(0) - 8 = -8
So the y-intercept of the parabola is (0,-8).

To find the vertex we use: 

How to find quadratic equation from graph without vertex

How to find quadratic equation from graph without vertex
and to find k, we plug in -1 in for x:
k = (-1)2 + 2(-1) - 8
k = 1 - 2 - 8 = -9
The vertex of this parabola is at (-1, -9)

How to find quadratic equation from graph without vertex


Example 2)

Graph y = -3x2 + 3

In this problem a = -3, b = 0 and c = 3.

Since "a" is negative this parabola is going to open downward (upside down U shape).

To find the x-intercepts we plug in 0 for y:
0 = -3x2 + 3 (this equation factors)
0 = -3(x2 - 1)
0 = -3(x - 1)(x + 1) and since -3 can not equal zero:
x = 1 or x = -1
The x-intercepts are: (1,0) and (-1,0)

The y-intercept is found by plugging 0 for x:
y = -3(0)2 + 3 = 3
So, the y-intercept is at (0,3).

And to find the vertex:

How to find quadratic equation from graph without vertex

k

= -3(0)2 + 3 = 3
So the vertex is at (0,3).

Notice that in this problem the vertex and the y-intercept are the same point.

How to find quadratic equation from graph without vertex


Example 3)

Graph y = x2 + 4x + 7  

a = 1, b = 4, and c = 7

Since a 0 the parabola opens up (is U shaped).

To find the x -intercept we plug in 0 for y:
0 = x2 + 4x + 7 (this expression does not factor so we have to use the quadratic formula)

How to find quadratic equation from graph without vertex

Since the roots are imaginary the parabola has no x-intercepts.

We find the y-intercepts by plugging in 0 for x:
y = 02 + 4(0) + 7 = 7
The y-intercept is (0,7).

The vertex:

How to find quadratic equation from graph without vertex

So the vertex is at (-2, 3).

How to find quadratic equation from graph without vertex

How do I find a quadratic equation given 2 points and no vertex?

How do i find the equation of a parabola given 2 points and the axis of symmetry, but no vertex?.
Using the vertex form of a parabola f(x) = a(x - h)2 + k where (h,k) is the vertex of the parabola..
The axis of symmetry is x = 0 so h also equals 0..
a = 1..
Substituting the a value into the first equation of the linear system:.

How do you find the vertex without vertex form?

To find the vertex (h, k) of a parabola that is in intercept form y = a(x - p) (x - q): Use h = (p + q) / 2 for finding h. Substitute x = h in the equation of parabola to find k.

How do you find the vertex in a quadratic function graph?

We find the vertex of a quadratic equation with the following steps:.
Get the equation in the form y = ax2 + bx + c..
Calculate -b / 2a. This is the x-coordinate of the vertex..
To find the y-coordinate of the vertex, simply plug the value of -b / 2a into the equation for x and solve for y..