Question

How does a change in the value of \(c\) change the graph of \(y=a x^{2}+b x+c ?\)

Short answer

Changing the value of \(c\) in the quadratic function \(y = ax^2 + bx + c\) moves the graph vertically. An increase in \(c\) shifts the graph upwards, while a decrease in \(c\) shifts the graph downwards.

Step-by-step solution

Understanding the Quadratic Function

First, let's take a look at a standard form quadratic function, \(y = ax^2 + bx + c\). Each variable in this equation has a unique role: \(a\) dictates the direction and the width of the parabola, \(b\) determines the line of symmetry, and \(c\) provides the y-intercept of the graph.

Analyzing the Role of \(c\)

Now, let's focus on \(c\), which represents the y-intercept, the point where the graph crosses the y-axis. When you change the value of \(c\), you're effectively moving the graph vertically up or down without affecting the shape or orientation of the graph. So, if \(c\) increases, the graph moves up. If \(c\) decreases, the graph moves down.

Visualizing the Change

To visualize this, plot different graphs by changing the value of \(c\) while keeping \(a\) and \(b\) constant. For example, compare the graphs of \(y = x^2\), \(y = x^2 + 1\), and \(y = x^2 - 1\). You'll see that each change in \(c\) results in the graph shifting up or down.