The Science That Draws Necessary Conclusions

Quadratic equations are equations that have the variable raised to the second power. They are in the form of .

The graph of a quadratic equation looks like a cross between a U and a V. There are two roots (where the funtion returns 0 also called a zero); they can be imaginary and both can be the same. In the graph above, the roots are 0.

A common way to find the roots of a quadratic equation is with the quadratic formula. It is where x is the root. It has two solutions, one using the plus sign and the other using the subtraction sign. The nature of the roots can be found with the discriminant. The discriminant is the part from under the square root, . If it is negative, there are two imaginary zeros. If the discriminant is 0, there is one real roots. If the discriminant is positive, there are two real roots.

There is more too the quadratic function, but it can be explained along with all other polynomial equations.

A linear function is a function whose graph is a line. They are written in three forms: slope-intercept form, point-slope form, and general form.

Slope-intercept is a common way to write a linear function. Its form is . The graph of is shown in the image above. The variables m and b modify this graph.

m is the slope of the line (how steep the line is). In , m is equal to 1. As a result, one point can be reached from another point by moving up 1 unit and to the right 1 unit. The function has a slope of 2 so it will move up 2 units and to the right 1 unit. The function has a slope of so it will move up 1 unit and to the right 2 units. When m is negative, the line will be moving down rather than up.

b is the y-intercept of the line (where the line intersects the y-axis). The y-intercept of is 0. As a result, the line crosses the y-axis at the origin. The line crosses the y-axis at -3.

Point-slope form is . This can be written as a function as (y become f(x)). and are the coordinates of a point on the graph. This can be useful if the slope and a point on the graph is known. For example, a line has a slope of 3 and passes through the point (3, 5). with the point and slope plugged in becomes . After distributing the 3 and adding the -9 to the 5 gives the graphable slope-point form .

General form is . It is less commonly used and at the moment not that important. It can be converted to point-slope form by changing it to . The slope is and the y-intercept is .

The graph above can be written as (slope-point), (point-slope), or as (general form).

A relation is composed of ordered pairs of numbers. The set of the first numbers is the domain and the set of the second values is the range of the relation. The first number is called the abscissa and the second number is called the ordinate.

The ordered pairs can be graphed using the Cartesian coordinate system. There are two axes in the coordinate system, the x and y axis. The x-axis is horizontal and the y-axis is vertical. The point where the lines cross is called the origin. That point is (0, 0). The further a point is toward the right, the larger the abscissa is. Moving toward the left from the origin goes negative. The further the point is upward, the larger the ordinate of the point is. The abscissa can be called the x value and the ordinate can be called the y value.

A function is a special relation where each abscissa has no more than one ordinate. This means that any input will always yield the same output. Note that different inputs may share the same output. The input is the independent variable x and the output is the dependent variable y.

The vertical line test tests whether a relation is a function. If a vertical line can be placed on a graph and it crosses through two or more points, then the line is not a function. The graph of a circle is not a function because a vertical line can be drawn that passes through more than one point. The picture of the graph on the right is a function because no matter where a vertical line is placed, it will only pass through one point. Each x value has only one y value.

A functions is written as . f(x) is the same as the y coordinate. So if , then and (3, 9) is a point of the graph of f(x). There are numerous functions to learn, but luckily there are patterns.