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How To Find the Vertex of a Parabola

find vertex of parabola: Graph of a quadratic equation

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.

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What Is the Pythagorean Theorem and When Is It Used?

What is the Pythagorean theorem: Graph of how the sum of the squares creates the right triangle ABC

What is the Pythagorean theorem? It’s a trigonometry equation used to find the length of one side of a right triangle. Though similar concepts had been discovered by the Babylonians, Greek Mathematician Pythagoras was the first person to come up with a geometric proof about how the sum of the squares of the lengths can determine the side lengths of a right triangle.

Pythagoras determined that when three squares are arranged so that they form a right angle triangle, the largest of the three squares must have the same area as the other two squares combined. In the picture below, you can see how the sum of the squares creates the right triangle ABC.

This realization about the area of the squares led to the Pythagoras theorem:

What is the Pythagorean theorem: a2+b2=c2 formula

Squares are different from other parallelograms and trapezoids because all their sides are equal lengths. So since squares are made up of four equal sides, you can see that each individual square makes up a side of the right triangle.

The length of the largest square, which we'll call length c, is the length of the hypotenuse. (The hypotenuse is the longest side of a right triangle.) The smaller squares make up the other two sides of the right triangle.

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How To Find the Focus of a Parabola: Your Easy Guide

how to find the focus of a parabola: Diagram of a parabola

If you want to know how to find the focus of a parabola, you first need to define what a parabola is. A parabola is a curved figure where any point on the curve is equal distance from a fixed point (called the focus) and a fixed straight line (called the directrix).

Let’s identify the parts of a parabolic function. In the graph above, you see a given line that intersects the directrix at a 90-degree angle. This straight line is called the axis of symmetry. The point that is marked C, signifying where the parabola opens, is called the vertex. The vertex is always midway between the focus and directrix of a parabola.

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Here’s How To Find the Hypotenuse of a Right Triangle

how to find hypotenuse: Diagram of a triangle with an unknown hypotenuse

The Pythagorean Theorem is an important theorem, as it comes up often in high school math. It makes the question of how to find the hypotenuse of a right triangle is easy to answer.

Plus, we can use it to find the base of a special right triangle and apply the converse of the Pythagorean Theorem. Let’s review this basic but foundational concept in trigonometry and how it relates to finding the hypotenuse of a right triangle.

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What To Remember When Graphing Linear Equations

graphing linear equations: Cartesian graph

Graphing linear equations is key to helping you with making observations or insights about the equation. Before we look at graphing linear equations, let's review graphing basics.

The image above is a Cartesian graph. The x-axis goes left to right, and the x-intercept is wherever a straight line crosses over or intersects this horizontal axis. The y-axis goes up and down, and the y-intercept is wherever a straight line crosses over or intersects this vertical axis. Think of this as a rectangular coordinate system formed with evenly spaced perpendicular lines. Each ordered pair of coordinates (x,y) denotes a singular point on the graph.

The graph of a linear equation can be drawn by finding which x values and y values solve for the equation of the line, meaning those values balance the equation. For example, (2, 5) is a valid graph point for the equation graphing linear equations: y=2x+1 formula, because substituting 2 for x and 5 for y in the equation gives 5 = 2(2)+1, or 5=5. (0, 0) is not a solution because 0≠0+1, or 0≠1, and therefore is not a point on the line. Once you have collected several points, you can graph your equation.

Find the x-coordinate of a point on the graph by counting the number of spaces along the x-axis in accordance with the first value. Then, find the y-coordinate by looking at the second number and counting that number of spaces up or down the y-axis. For example, the point (2,3) would be two spaces to the right and three spaces up.

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