C-Value: Definition & Explanation

Instructor: Sarah Spitzig

Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada. She has recently earned a Master's degree.

The standard form of a parabola is y = ax^2 + bx + c where a, b, and c are parameters on the equation. In this lesson, you will learn how the c-value affects the graph of a parabola.

Definitions

Parabola

A parabola is a curved, U-shaped graph that is symmetrical, which means that any point on the graph has a mirror image on the other side. The equation will always have an x squared term and will open either up or down. These equations are often used to approximate the path of a projectile such as a ball or rocket.

Standard Form of a Parabola

There are many forms of the equation of a parabola, such as vertex form, factored form, and standard form. In this lesson, we will focus on standard form, which is --

y = ax^2 + bx + c

a, b, and, c values

The a, b, and c values are parameters on the graph of the equation in standard form. In other words, as a, b, and c change, the graph changes as well. Remember that a must be a non-zero real number, while b and c can be any real number. (b and c could be zero as well.)

Graph of a parabola

The graph of a parabola is a U-shaped curve that either opens up or down. Whether it opens up or down and the width (narrow or fat) depends on the value of a.

Vertex

The vertex of the graph is the highest point if the graph opens down and the lowest point if the graph opens up. When the vertex is the highest point on the graph, it is called a maximum. When the vertex is the lowest point, it is called a minimum.

c-value

The c-value can also be called the y-intercept of the parabola. Algebraically, this is where the x-value is zero and graphically, this is where the graph intersects the y-axis.

Graphs of Parabolas

The graph of a parabola that opens down looks like this. The c-value is where the graph intersects the y-axis. In this graph, the c-value is -1 and its vertex is the highest point on the graph known as a maximum.

graph of a parabola

The graph of a parabola that opens up looks like this. The c-value is where the graph intersects the y-axis. In this graph, the c-value is -1 and its vertex is the lowest point on the graph known as a minimum.

graph of a parabola

Changing the C-value

Changing the c-value changes where the graph intersects the y-axis. For example, the graphs of the following equations would look almost the same. They would have the same width, the same vertex, and would both open in the same direction.

y = 2x^2 + 4x - 3 and y = 2x^2 + 4x + 7

The only difference is that one equation intersects y at -3 and the other intersects y at 7.

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