y=x^2-7
We want to draw the graph of the given function, which has the form y=ax^2+k, where a is different from 0 and k is a constant. To do so, we will follow four steps.
Find and plot three points on one side of the axis of symmetry.
Plot the corresponding points on the other side of the axis of symmetry.
Sketch the curve.
Step 1
The vertex of anyparabola with an equation following the form y=ax^2+k is always (0,k). Since the axis of symmetry is the vertical line through the vertex, its equation is x=0.
Step 2
Now we need to find and plot any three points on one side of the axis of symmetry. For simplicity, we will plot the points whose x-coordinates are 1, 2, and 3. Let's use a table to find the y-coordinates of these points.
x
x^2-7
y=x^2-7
1
1^2-7
-6
2
2^2-7
-3
3
3^2-7
2
We found that the points ( 1, - 6), ( 2, -3), and ( 3, 2), are on the curve. Let's plot them!
Step 3
We will continue by plotting the corresponding points on the other side of the axis of symmetry. This means that we will reflect the three points we found across the line x=0.
Note that the y-coordinates do not change, but the x-coordinates have opposite signs.
Step 4
Finally, we connect the points with a smooth curve. We do not need a straight edge for this!
Once we have graphed the function, we can answer the question. The roots of the function are the x-coordinates for which the function equals 0. This means that we need to find points of intersection of the graph of the function and the x-axis.
We can see that the graph crosses the x-axis twice. Therefore, the graph has two roots.
b To find the roots of the function, we need to calculate the x-coordinates of the points we found in Part A. Since we cannot precisely tell the coordinates from the graph, we will find them algebraically. Recall that such points have y-coordinates equal to 0.
0=x^2-7
Let's isolate the variable on one side of the equation and take the square root of each side. Keep in mind that we need to consider the positive and negative solutions.
