There are two methods to find the vertex of a quadratic function. Problems on Vertex and Interecpts of Quadratic Functionsįind the vertex, the x and y intercepts and the axis of symmetry of the graph of the quadratic equation given by f(x) = - x 2 - 2 x + 3 If the graph of f has x intercepts at x 1 and x 2 then x 1 and x 2 are solutions to the equation f(x) = 0 so that f may be given by It can be easily shown that h = (x 1 + x 2) / 2 The x intercepts (if any) of the graph are the point of intersection of the graph of f and the x axis and are therefore the solutions to the equation a x 2 + b x + c = 0 whose solutions are given by the quadratic formulas as follows: The axis of symmetry of the parabola is a vertical line given by the equation: x = h. The quadratic function of the parabola whose axis is vertical and whose vertex is at the point ( h, k ) is given by The vertex has coordinates ( h, k) where h = - b / 2a When the coefficient a is positive the vertex is the lowest point in the parabola that opens upward and when it is negative, the vertex is the highest point in the parabola that opens downward. Is a vertical parabola with axis of symmetry parallel to the y axis and has a vertex V with coordinates (h, k), x - intercepts when they exist and a y - intercept as shown below in the graph. The graph of a quadratic function of the form Review Vertex and Intercepts of a Quadratic Functions Try out the practice question to test your understanding.Vertex and Intercepts of Quadratic FunctionsĪ set of problems related to the vertex, the x and y interceptsĪnd the axis of quadratic functions are presented along with analytical solutions and graphical interpretations.Ī calculator to findthe vertex and intercepts of quadratic functions may be used to check calculations. Hence, the coordinates of the vertex of this quadratic equation is (1, 1). Hence, the y-coordinate of the vertex is 1. Now, we can use this x-coordinate to find the y-coordinate of the vertex. Let's change back to the original equation. With this, the vertex has the x-coordinate of 1. Knowing this, we can substitute 'b' with 2 and substitute 'a' with -1. Hence, we can see that, 'a' is equals to -1, ‘b’ is equals to 2, and 'c' is equals to 0. To find the values of 'a' and 'b', we can compare this equation with the general equation, y = ax^2 + bx +c.įor easier comparison, we can rewrite this quadratic equation as, y = -1x^2 +4x + 0. Let's find the coordinates of this vertex.Īgain, we start with the formula for the x-coordinate of the vertex of a quadratic equation, x = -b/2a. Now, the vertex is located at the highest point on the graph. The graph for this equation is shown here. Consider the quadratic equation, y = -x^2 +2x. Hence, the vertex of the equation has the coordinates of (2, -5). Hence, the vertex has y-coordinate of -5. Adding -16 with 3, gives -13.Īgain, we can substitute x with 2. Now, we can use the x–coordinate to find the y-coordinate of the vertex. With this, we know that the vertex of the quadratic equation has the x-coordinate of 2. 8 divide by 4, gives 2įinally, we have x equals to 2. Knowing this, we can substitute 'b', with -8, and substitute 'a', with 2. To find out the values of 'a', 'b' and 'c', we can rewrite this equation as, y = 2x^2 + (-8)x +3.īy comparing this equation with the general equation, we can see that, 'a' is equals to 2, 'b' is equals to -8, and 'c' is equals to 3. Let's see an example on using this formula, by using this equation, y = 2x^2 -8x +3. Where, 'a', 'b' and 'c' are the coefficients for each term respectively. The general equation of a quadratic equation is given as, y = ax^2 +bx, +c. Now, what are 'a' and 'b'? Let's find out. Now, there is a formula to calculate the x-coordinate of the vertex of a quadratic equation. Hence, this point is the vertex of a quadratic equation.Īlso, if you have a quadratic graph as shown here, the lowest point, is also the vertex of the quadratic equation, y = 2x^2 -8x +3. Notice that, this is the highest point on the graph. Now, this is the graph of the quadratic equation, y= -x^2 +2x. In this lesson, we will learn about the vertex of a quadratic equation.
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