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Here are two equations that define quadratic functions.
The graph of passes through and , as shown on the coordinate plane.
Find the coordinates of another point on the graph of . Explain or show your reasoning. Then use the points to sketch and label the graph.
Priya says, "Once I know that the vertex is , I can find out, without graphing, whether the vertex is the maximum or the minimum of function . I can just compare the coordinates of the vertex with the coordinates of a point on either side of it."
Complete the table, and then explain how Priya might have reasoned about whether the vertex is the minimum or maximum.
| 3 | 4 | 5 | |
| 10 |
Not surprisingly, vertex form is especially helpful for finding the vertex of a graph of a quadratic function. For example, we can tell that the function, , given by has a vertex at .
We also noticed that, when the squared expression has a positive coefficient, the graph opens upward. This means that the vertex, , represents the minimum function value, .
Parabola in x y plane, origin O. X axis 0 to 6, by 1’s. Y axis 0 to 10, by 2s. Parabola opens upward with vertex at 3 comma 1. Left side of parabola crosses the y axis at 10.
But why does function take on its minimum value when is 3?
Here is one way to explain it: When , the squared term equals 0, because . When is any other value besides 3, the squared term is a positive number greater than 0. (Squaring any number results in a positive number.) This means that the output when will always be greater than the output when , so function has a minimum value at .
This table shows some values of the function for some values of . Notice that the output is the least when , and it increases both as increases and as it decreases.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 10 | 5 | 2 | 1 | 2 | 5 | 10 |
The squared term sometimes has a negative coefficient, for instance in . The value that makes equal 0 is -4, because . Any other value makes greater than 0. But when is multiplied by a negative number like -2, the resulting expression, , ends up being negative. This means that the output when will always be less than the output when , so function has its maximum value when .
Remember that we can find the -intercept of the graph representing any function that we have seen. The -coordinate of the -intercept is the value of the function when . If is defined by , then the -intercept is because . Its vertex is at . Another point on the graph with the same -coordinate is located the same horizontal distance from the vertex but on the other side.
Parabola in x y plane, origin O. X axis negative 4 to 1, by 1’s. Y axis negative 8 to 2, by 2s. Parabola opens upward with vertex at negative 1 comma negative 5. Other points on the parabola given are negative 2 comma negative 4 and 0 comma negative 4.