Quadratic Functions in Standard Form
Convert to a hyperbola to standard form to find foci, vertices, center and asymptotesand
Quadratic functions are some of the most important algebraic functions and they need to be thoroughly understood in any modern high school algebra course. The properties of their graphs such as vertex and x and y intercepts are explored interactively using an html5 applet. The exploration is carried by changing values of 3 coefficients a , b and c included in the definition of f x. Once you finish the present tutorial, you may want to go through tutorials on quadratic functions , graphing quadratic functions and Solver to Analyze and Graph a Quadratic Function There are two more pages on quadratic functions whose links are shown below. Interactive Tutorial 1 Explore quadratic functions interactively using an html5 applet shown below; press "draw' button to start.
Index of lessons Print this page print-friendly version Find local tutors. For graphing, the leading coefficient " a " indicates how "fat" or how "skinny" the parabola will be. Also, if a is negative, then the parabola is upside-down. As you can see, as the leading coefficient goes from very negative to slightly negative to zero not really a quadratic to slightly positive to very positive, the parabola goes from skinny upside-down to fat upside-down to a straight line called a "degenerate" parabola to a fat right-side-up to a skinny right-side-up. There is a simple, if slightly "dumb", way to remember the difference between right-side-up parabolas and upside-down parabolas:. This can be useful information: If, for instance, you have an equation where a is negative, but you're somehow coming up with plot points that make it look like the quadratic is right-side-up, then you will know that you need to go back and check your work, because something is wrong.
I f you are given the solutions of an equation, you can find an equation by working backward. This equation can be solved by factoring. There are additional methods for creating equations from graphs that will be discussed in Algebra 2. Can you determine the equation of a quadratic given its solutions? Let's see what's possible. Given the naked parabolic graph at the right, you are told that the x -intercepts and y -intercept are integer values.
The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape. The picture below shows three graphs, and they are all parabolas. All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola. You know that two points determine a line. This means that if you are given any two points in the plane, then there is one and only one line that contains both points.
The graph of a quadratic function is a parabola, and its parts provide valuable information about the function. The graph of a quadratic function is a U-shaped curve called a parabola. This shape is shown below. This is shown below. Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane.
A quadratic function is a polynomial function of degree 2 which can be written in the general form,. Note that the graph is indeed a function as it passes the vertical line test. When graphing parabolas, we want to include certain special points in the graph. The y -intercept is the point where the graph intersects the y -axis. The x -intercepts are the points where the graph intersects the x -axis. The vertex The point that defines the minimum or maximum of a parabola.
Graphing Quadratic Equations
9-1 Quadratic Graphs and Their Properties