![]() To find d and f, we need to analyze the original equation. This makes sense because if two numbers, a and b, are such that ab=0, the either a=0, b=0 or both. If a quadratic equation can be so reduced, the graph of f(x) will cross the x-axis at the points (-d, 0) and (-f, 0). Some quadratic formulae of the form f(x)=ax 2+bx+c are reducible to the form (x+d)(x+f), where d and f are whole numbers. The simplest method is factoring, but we can also use the quadratic formula for x-intercepts that are not whole numbers. There are several ways to find these x-values. Otherwise, the function will have two x-intercepts. Likewise, if the vertex of the function lies on the x-axis, the function will only have one x-intercept. For example, if the vertex of the function is above the x-axis and the function points upwards or the vertex is below the x-axis and the function points downwards, the parabola will not cross the x-axis. Note that not every parabolic function has an x-intercept. These will in turn be the x-values of the x-intercepts of the parabolas modeling f(x). That is, if we plug the value of x into the equation for the quadratic, it simplifies to 0. The “solution” of a quadratic equation is an x-value for which f(x)=0. In order to find these, however, we have to know how to solve quadratic equations. This formula for finding the vertex may seem complicated, but it is actually related to certain points on the parabola. Note that, when we just have the function x 2, the vertex is the origin, (0, 0). The vertex of a parabola is the point ( -b/ 2a, f( -b/ 2a)). The vertex of a parabola is the lowest point of an upward pointing parabola and the highest point of a downward pointing parabola. This means we can find the y-intercept of a quadratic function by evaluating the function when x=0.įor a quadratic function of the form ax 2+bx+c, we get a(0) 2+b(0)+c=0+0+c. The y-intercept of a parabola or any function is the point where x=0. Therefore, since the vertex has coordinates ( -b/ 2a, f( -b/ 2a)). This line goes right through the vertex of the function. Not every quadratic function is even because some have an x term, but every quadratic function does have a line of symmetry. This means the graph of the function on one side is the mirror image of the graph of the function on the other side. This makes sense because (-x) 2=x 2.Įven functions have a line of symmetry equal to x=0, the y-axis. This means that the function has the same value for x and -x. This makes sense because we reflect functions over the x-axis by multiplying them by a negative. If a is negative, the parabola points downwards. If a is positive, the parabola turns upwards. We know whether the parabola of our quadratic function will turn upwards or downwards based on the value of a. Note that concave parabolas may also be known as concave up and convex parabolas may also be known as concave down. On the other hand, the y-values of the latter will extend to negative infinity as x goes towards positive or negative infinity. Solutions of the former will extend to positive infinity as x goes towards positive or negative infinity. We call parabolas that curve upwards “concave” and parabolas that curve downward convex. ![]() ShapeĮvery parabola will turn upwards like a smiley face or turn downwards like a frown. In particular, we are interested in the vertex, the y-intercept, the x-intercept(s), and the general shape of the graph. When a is not equal to 0, however, we need to use the values of a, b, and c to tell us about the graph. ![]() If the value of a is 0, then we simply have a linear function and can graph it like any other linear function. Recall that quadratic functions have the form ax 2+bx+c, where a, b, and c are real numbers. Make sure to review these concepts before moving forward. Graphing quadratics requires a solid understanding of coordinate geometry and graphing. Mathematically, such functions are called concave and convex or “concave up” and “concave down.” The parabolic shape is often likened to a smiley face or a frowning face. These functions will generally form a parabola. Graphing quadratic functions models an x 2 function in two-dimensional space.
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