The vertex form of a quadratic function is \( f\left( x \right) = a{(x - h)}^{2} + k. \)
The ordered pair \( (h,k) \) represents the vertex of the parabola. This is the lowest or highest point on the graph of the function. The value of a determines which way the parabola opens.
Let's Watch
Watch the video instructor below as he explains how the values of a, h, and k affect the graph of the parent function \( f\left( x \right) = x^{2} \). This information will be helpful to you as you begin to graph the transformations of a quadratic function.
You may want to follow along using the study guide from earlier in the lesson.
You should notice that the vertex form of the quadratic has three numbers that are used to transform the parent function (pause). The numbers are in the position a, h, and k. The a-value, which is the same as the a-value in standard form, identifies the vertical stretch or compression of the parabola, and whether the parabola opens upward or downward. If a is positive, the graph opens up, and if a is negative, the graph opens down. When the absolute value of a is greater than one, the parent graph will stretch taller by a factor of a, and if the absolute value of a is between zero and one, the graph will compress shorter.
The values of h and k determine the x and y coordinate of the vertex which is the top or bottom of the parabola. Notice that in this form, the opposite of h is indicated in the function, so we will use the opposite of this number when identifying the vertex.
Practice identifying the key graph components from the vertex form of the function by completing the activity below. For each function, identify the vertex and which way the parabola will open. Then state if the parabola will be stretched taller than or compressed shorter than the parent function, \( f\left( x \right) = x^{2} \). Click the function to check your answer.
