04-3a+Solving+Quadratics+by+Graphing+and+Factoring


 * Standard & Assessment Anchor || M11.D.2.1.5 Solve quadratic equations using factoring (integers only – not including completing the square or the Quadratic Formula). ||
 * Objective || We will learn to solve quadratic equations by graphing and factoring, and to recognize special patterns. ||
 * Objective || We will learn to solve quadratic equations by graphing and factoring, and to recognize special patterns. ||

Why do we study quadratic equations? Good question!

If you are asking that question, you probably have not yet taken Physics, and don't really know much about the effect that gravity has on a projectile (that's an object that has been launched from a stated position). If you have taken Physics, you will recognize the acceleration due to gravity, the datum plane, and other physics related terms that we just might reference in this lesson.

We can apply a quadratic function to the game of soccer, which most people have at least a passing familiarity with. We can model the height of the ball during its flight at a time t by using the function h(t) = -16t 2 + 32t. In this function, the ball is kicked from the ground, so its initial height is zero, and when it again hits the ground its final height is also zero. While the ball is in the air, its path traces a parabola. Let's look at the graph of this function:



From the graph we see that the ball is in the air for 2 seconds before it hits the ground, and reaches a maximum height of 16 feet.

We call the values of x where the value of the function is zero, coincidentally, the zeroes of the function. They are also known as the x-intercepts.

When we studied linear functions, we noticed that the graph of the line crossed the x-axis in one place, so a linear function has one zero, or one x-intercept. A quadratic function can have up to two zeroes, which are always symmetric about the axis of symmetry.

For relatively simple quadratic functions, we can find the zeroes of the function by drawing a graph or generating a table of values.

Let's look at the function f(x) = x 2 - 6x + 8

We know that the function opens upward since the coefficient of the square term, a, is greater than zero. The y-intercept is 8 because c = 8.

Let's find the vertex using the formula:



Now let's evaluate the function at the vertex:



The vertex is at (3, -1)

We can plot the vertex and the y-intercept. Using symmetry and generating a table of values, we can find additional points.


 * x || -1 || 0 || 6 || 1 || 5 || 2 || 4 ||
 * y || 3 || 8 || 8 || 3 || 3 || 0 || 0 ||



The table and the graph indicate that the zeroes are 2 and 4.

We could also have accomplished the same task by using the graphing calculator; enter the function at Y=, graph, then view the table of values.

and

Proceed to Finding Zeroes Algebraically