There are several mathematical functions that use vertices. A geometric figure has several vertices, a system of inequalities has one or more vertices, and a parabola or quadratic equation also has vertices. How to find vertices depends on the situation, but here are a few things you should know about finding vertices in each scenario.
Step
Method 1 of 5: Finding the Number of Vertexes in a Shape
Step 1. Learn Euler's Formula
Euler's formula, as referred to in geometry or graphs, states that for any shape that is not tangent to itself, the number of edges plus the number of vertices, minus the number of edges, will always equal two.
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If written in the form of an equation, the formula looks like this: F + V - E = 2
- F refers to the number of sides.
- V refers to the number of vertices, or vertices
- E refers to the number of ribs
Step 2. Change the formula to find the number of vertices
If you know the number of sides and edges that a shape has, you can quickly calculate the number of vertices by using Euler's Formula. Subtract F from both sides of the equation and add E on both sides, leaving V on one side.
V = 2 - F + E
Step 3. Enter the known numbers and solve
All you need to do at this point is plug the number of sides and edges into the equation before adding or subtracting normally. The answer you get is the number of vertices and thus solves the problem.
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Example: For a rectangle that has 6 sides and 12 edges…
- V = 2 - F + E
- V = 2 - 6 + 12
- V = -4 + 12
- V = 8
Method 2 of 5: Finding Vertexes in a System of Linear Inequality
Step 1. Draw the solution of the system of linear inequalities
In some instances, drawing the solutions of all the inequalities in the system can visually show some, or even all of the vertices. However, if you can't, you need to find the vertex algebraically.
If you're using a graphing calculator to draw the inequality, you can swipe up on the screen to the vertex point and find its coordinates that way
Step 2. Turn the inequality into an equation
To solve a system of inequalities, you need to temporarily convert the inequalities into equations in order to find the value of x and y.
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Example: For a system of inequalities:
- y < x
- y > -x + 4
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Change the inequality to:
- y = x
- y > -x + 4
Step 3. Substitution of one variable to another variable
Although there are other ways to solve x and y, substitution is often the easiest way. Enter value y from one equation into another, which means "substituting" y into another equation with the value of x.
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Example: If:
- y = x
- y = -x + 4
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So y = -x + 4 can be written as:
x = -x + 4
Step 4. Solve for the first variable
Now that you have only one variable in the equation, you can easily solve for the variable, x, as in other equations: by adding, subtracting, dividing and multiplying.
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Example: x = -x + 4
- x + x = -x + x + 4
- 2x = 4
- 2x / 2 = 4 / 2
- x = 2
Step 5. Solve for the remaining variables
Enter a new value for x into the original equation to find the value of y.
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Example: y = x
y = 2
Step 6. Define the vertices
The vertex is the coordinate containing the value x and y that you just discovered.
Example: (2, 2)
Method 3 of 5: Finding the Vertex on a Parabola Using the Axis of Symmetry
Step 1. Factor the equation
Rewrite the quadratic equation in factor form. There are several ways to factor a quadratic equation, but when you're done, you'll have two groups in brackets, which when you multiply them together, you'll get the original equation.
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Example: (using parsing)
- 3x2 - 6x - 45
- Outputs the same factor: 3 (x2 - 2x - 15)
- Multiplying the coefficients a and c: 1 * -15 = -15
- Finds two numbers which when multiplied equals -15 and whose sum equals the value b, -2; 3 * -5 = -15; 3 - 5 = -2
- Substitute the two values into the equation 'ax2 + kx + hx + c: 3(x2 + 3x - 5x - 15)
- Factoring by grouping: f(x) = 3 * (x + 3) * (x - 5)
Step 2. Find the x-intercept of the equation
When the function x, f(x), equals 0, the parabola intersects the x-axis. This will happen when any factor is equal to 0.
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Example: 3 * (x + 3) * (x - 5) = 0
- +3 = 0
- - 5 = 0
- = -3; = 5
- So, the roots are: (-3, 0) and (5, 0)
Step 3. Find the midpoint
The axis of symmetry of the equation will lie exactly halfway between the two roots of the equation. You have to know the axis of symmetry because the vertices lie there.
Example: x = 1; this value is exactly in the middle of -3 and 5
Step 4. Plug the value of x into the original equation
Plug the x-value of the axis of symmetry into the equation of the parabola. The y value will be the y value of the vertex.
Example: y = 3x2 - 6x - 45 = 3(1)2 - 6(1) - 45 = -48
Step 5. Write down the vertex points
Up to this point, the last calculated values of x and y will give the coordinates of the vertex.
Example: (1, -48)
Method 4 of 5: Finding the Vertex on a Parabola by Completing the Square
Step 1. Rewrite the original equation in vertex form
The "vertex" form is an equation written in the form y = a(x - h)^2 + k, and the vertex point is (h,k). The original quadratic equation must be rewritten in this form, and for that, you must complete the square.
Example: y = -x^2 - 8x - 15
Step 2. Get the coefficient a
Remove the first coefficient, a from the first two coefficients of the equation. Leave the last coefficient c at this point.
Example: -1 (x^2 + 8x) - 15
Step 3. Find the third constant inside the brackets
The third constant must be enclosed in brackets so that the values in the brackets form a perfect square. This new constant is equal to the square of the half coefficient in the middle.
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Example: 8 / 2 = 4; 4 * 4 = 16; so that,
- -1(x^2 + 8x + 16)
- Remember that the processes performed inside the brackets must also be performed outside the brackets:
- y = -1(x^2 + 8x + 16) - 15 + 16
Step 4. Simplify the equation
Since the shape inside the brackets is now a perfect square, you can simplify the shape inside the brackets into factored form. Simultaneously, you can add or subtract values outside the brackets.
Example: y = -1(x + 4)^2 + 1
Step 5. Find the coordinates based on the vertex equation
Recall that the vertex form of the equation is y = a(x - h)^2 + k, with (h,k) which is the coordinates of the vertices. Now you have complete information to enter values into h and k and solve the problem.
- k = 1
- h = -4
- Then, the vertex of the equation can be found at: (-4, 1)
Method 5 of 5: Finding the Vertex on a Parabola using a Simple Formula
Step 1. Find the x value of the vertex directly
When the equation of the parabola is written in the form y = ax^2 + bx + c, x of the vertex can be found by the formula x = -b / 2a. Just plug the a and b values from the equation into the formula to find x.
- Example: y = -x^2 - 8x - 15
- x = -b / 2a = -(-8)/(2*(-1)) = 8/(-2) = -4
- x = -4
Step 2. Plug this value into the original equation
Plugging the value of x into the equation, you can find y. The y value will be the y value of the vertex coordinates.
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Example: y = -x^2 - 8x - 15 = -(-4)^2 - 8(-4) - 15 = -(16) - (-32) - 15 = -16 + 32 - 15 = 1
y = 1
Step 3. Write down the coordinates of the vertices
The x and y values you get are the coordinates of the vertex point.
