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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.

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 find vertex of parabola: Standard form of a quadratic function.

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

find vertex of parabola: Axis of symmetry formula

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.

find vertex of parabola: Axis of symmetry formula

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

Quadratic equation

Quadratic equation

Quadratic equation

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.

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