The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.
A parabola either opens upward or downward. This is called its direction.
At the vertex, the function changes from increasing to decreasing, or vice versa.
All parabolas are symmetric, meaning there exists a line that divides the graph into two mirror images. For quadratic functions, that line is always parallel to the -axis, and is called the axis of symmetry.
Depending on its rule, a parabola can intersect the -axis at or points. Since the function's value at an -intercept is always these points are called zeros, or sometimes roots.
For the quadratic function create a table of values to graph it. Then determine its direction, vertex, zeros, and axis of symmetry.
We'll plot these points on a coordinate plane.
We can start to see the left-hand side of the parabola. Let's add a few more -values to the table to determine a more complete shape.
We'll add these points to the coordinate system as well.
Looking at the points, we now see both sides of the parabola. We can connect the points with a smooth curve.
The graph can be used to describe the desired characteristics of the parabola.
Three quadratic functions are graphed in the coordinate plane.
For each graph, match it with the corresponding characteristics.
Instead of looking at each function separately, we'll look at the characteristics individually and summarize our findings in a table at the end.
First, let's consider the direction of the parabolas. We can see that and open upward, and that opens downward. The direction of a parabola determines whether the vertex is a minimum or a maximum. Thus, the vertices of and are minimums while the vertex of is a maximum.
The vertex for each graph is found at the minimum or maximum of the function. For the vertex lies at Similarly, 's vertex lies at , and has its vertex at
The axis of symmetry is the vertical line that intersects the vertex. Therefore, the axis of symmetry for graph is for it's and for it is
The -intercepts are found where the parabolas intercept the -axis. has the -intercept , has and has . One of the options, does not coincide with any graph.
The zeros are found where the parabolas intercept the -axis. Then, function has zeros and , which are two of the zeros given in the prompt. Neither function nor function has a zero at .
We summarize what we have learned about the characteristics of the three quadratic functions.
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