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Next, we determine the x-value of the vertex. Because there are no real solutions, there are no x-intercepts. Since the discriminant is negative, we conclude that there are no real solutions. Use the discriminant to determine the number and type of solutions.ī 2 − 4 a c = ( 4 ) 2 − 4 ( 2 ) ( 5 ) = 16 − 40 = − 24 To find the x-intercepts, set f ( x ) = 0. Here c = 5 and the y-intercept is (0, 5). Y - i n t e r c e p t : ( 0, 3 ) x - i n t e r c e p t s : ( − 3, 0 ) and ( 1, 0 ) V e r t e x : ( − 1, 4 ) E x t r a p o i n t : ( − 2, 3 )īecause the leading coefficient 2 is positive, we note that the parabola opens upward. To recap, the points that we have found are Step 5: Plot the points and sketch the graph. Choose x = − 2 and find the corresponding y-value. In this example, one other point will suffice. Ensure a good sampling on either side of the line of symmetry. Step 4: Determine extra points so that we have at least five points to plot. Substitute −1 into the original function to find the corresponding y-value.į ( x ) = − x 2 − 2 x + 3 f ( − 1 ) = − ( − 1 ) 2 − 2 ( − 1 ) + 3 = − 1 + 2 + 3 = 4 One way to do this is to first use x = − b 2 a to find the x-value of the vertex and then substitute this value in the function to find the corresponding y-value.
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Hence, there are two x-intercepts, ( − 3, 0 ) and ( 1, 0 ). Here where f ( x ) = 0, we obtain two solutions. 0 = ( x + 3 ) ( x − 1 ) S e t e a c h f a c t o r e q u a l t o z e r o. 0 = − x 2 − 2 x + 3 M u l t i p l y b o t h s i d e s b y − 1. To do this, set f ( x ) = 0 and solve for x.į ( x ) = − x 2 − 2 x + 3 S e t f ( x ) = 0. Step 2: Determine the x-intercepts if any. The steps for graphing a parabola are outlined in the following example. However, in this section we will find five points so that we can get a better approximation of the general shape. Generally three points determine a parabola. L i n e o f s y m m e t r y V e r t e x x = − b 2 a ( − b 2 a, f ( − b 2 a ) ) We can use the line of symmetry to find the the vertex. Therefore, the line of symmetry is the vertical line x = − b 2 a. To do this, we find the x-value midway between the x-intercepts by taking an average as follows: Using the fact that a parabola is symmetric, we can determine the vertical line of symmetry using the x-intercepts. X - i n t e r c e p t s ( − b − b 2 − 4 a c 2 a, 0 ) and ( − b + b 2 − 4 a c 2 a, 0 ) Therefore, the x-intercepts have this general form: Doing this, we have a 2 + b x + c = 0, which has general solutions given by the quadratic formula, x = − b ± b 2 − 4 a c 2 a. Next, recall that the x-intercepts, if they exist, can be found by setting f ( x ) = 0. In general, f ( 0 ) = a ( 0 ) 2 + b ( 0 ) + c = c, and we have Given a quadratic function f ( x ) = a x 2 + b x + c, find the y-intercept by evaluating the function where x = 0. Many of these techniques will be used extensively as we progress in our study of algebra. Guessing at the x-values of these special points is not practical therefore, we will develop techniques that will facilitate finding them. In addition, if the x-intercepts exist, then we will want to determine those as well. (also called the axis of symmetry A term used when referencing the line of symmetry.) is the vertical line through the vertex, about which the parabola is symmetric.įor any parabola, we will find the vertex and y-intercept. Lastly, the line of symmetry The vertical line through the vertex, x = − b 2 a, about which the parabola is symmetric. is the point that defines the minimum or maximum of the graph. The vertex The point that defines the minimum or maximum of a parabola. The x-intercepts are the points where the graph intersects the x-axis. The y-intercept is the point where the graph intersects the y-axis. When graphing parabolas, we want to include certain special points in the graph. Furthermore, the domain of this function consists of the set of all real numbers ( − ∞, ∞ ) and the range consists of the set of nonnegative numbers [ 0, ∞ ). Note that the graph is indeed a function as it passes the vertical line test. and is shared by the graphs of all quadratic functions. This general curved shape is called a parabola The U-shaped graph of any quadratic function defined by f ( x ) = a x 2 + b x + c, where a, b, and c are real numbers and a ≠ 0.