Let's graph the function first, then we can answer the questions from previous exercise.
A quadratic function has a shape of a . To graph this specific function, we can make a table of values. Let's find the output values which correspond to the values of x from the following set.
{ -2, -1, 0, 1, 2 }
To do so, we will substitute each value of x in our function.
|c|c|c|
x & x^2+3 & y
- 2 & ( - 2)^2+3 & 7
- 1 & ( - 1)^2+3 & 4
0 & ( 0)^2+3 & 3
1 & 1^2+3 & 4
2 & 2^2+3 & 7
Let's plot these points and draw a parabola to represent this function.
Now, let's investigate the characteristics of the function!
Shape
We can tell that the parabola is going up along the y-axis. Therefore, we can describe our parabola as opening up. This is because the coefficient of x^2 is positive.
Line of symmetry
A is the line at which we could fold the graph so that both sides of the fold would perfectly map onto each other. From the graph, we can tell that the following pairs of points have the same y-coordinate and are the same distance from the x-axis.
(-2,7), (2,7) and (-1,4), (1,4)
This means that the line of symmetry, in this case, is the y-axis! We can see it on the graph below.
Does this work for all parabolas? Yes and no. Yes, because all parabolas have the same basic shape and each parabola will have a line of symmetry. The line of symmetry will always go through the vertex. No, because the line of symmetry is not always the y-axis. Sometimes parabolas are shifted to the left or right.
Special points
As we can tell from our graph, the special point on our parabola is its vertex. This is the point through which the line of symmetry passes, and it represents the minimum or maximum value of the function. In our case it is the minimum value, y=3.
x- and y-intercepts
Our function has a y-intercept of 3. It does not have any x-intercepts, because it has been shifted up in the vertical direction.
The highest or lowest point
As we already mentioned, the lowest point on our graph is the vertex, which occurs on the y-axis at y=3.
