d Use your results from the previous parts and recall that the y-intersect equals the c-value for a function of the form y=ax^2+bx+c.
E
e Substitute any of the points shown in the graph given. If the equation holds true after doing it, then it corresponds to the graph.
F
f Review the steps taken. What would change if a were not known?
A
a 2
B
b x=-2.5
C
c b=5
D
d y=x^2+5x+2
E
e See solution.
F
f No, see solution.
Practice makes perfect
a Recall that the y-intersect is the y-coordinate of the point where the graph intersects the y-axis. Let's take a look at the graph given.
As we can see, the graph intersects the y-axis at the point (0, 2). Therefore, the y-intersect is 2.
b Since the points (-4,-2) and (-1,-2) have the same height, one should be the reflection of the other across the axis of symmetry (AOS). Hence, they should be at the same distance d from the AOS.
Therefore, the AOS will be the vertical line passing through the midpoint of the points (-4,- 2) and (-1 ,-2). We can calculate the x-coordinate of the midpoint by averaging the x-coordinates of these points.
Since the AOS is a vertical line, its equation is just x = - 2.5
c From Part B, we know that the equation for the AOS is x=- 2.5. As the exercise indicates, we can also find the equation for the AOS by using the formula shown below.
x = - b/2aFurthermore, if we compare the form of the function with the standard form of a quadratic function, we can identify the value of a.
ax^2+bx+c
1x^2+bx+c
Hence, for this case, a= 1. Then, by substituting this a-value and x=- 2.5 in the formula above, we can solve for b.
d From Part B we know that the y-intersect is 2. Recall that for a function of the form y=ax^2+bx+c, the y-intersect equals the c-value. Hence, c=2. Furthermore, from Part C we know that b=5. Then, we can substitute these values in the equation given in the exercise to find the equation for our function.
y&= x^2+bx+c
& 1cm⇓
y&= x^2+5x+2
e In Part F we found the equation for our quadratic function, y=x^2+5x+2. Now, we will verify if this is the correct equation by substituting any of the three points indicated in the exercise graph. For example, let's try with (-1 ,-2 ).
Since the equality holds true, we know that the equation we found is the correct one.
f Notice that the steps we followed in this exercise allowed us to find the equation for the AOS. Then, we used this result and the formula for the AOS shown below.
x = - b/2a
Since we knew the corresponding value for x and a, we could solve for b. However, if we had not known the value of a there would not be enough information to solve for b. For this reason, this method will not work if the value of a is not known.
