b Examining the graph, we see that it is a parabola which means the graph corresponds to a second degree equation. We also see that it has two x-intercepts, x=2 and x=5. With this information, we can write the equation in factored form.
y=a(x-2)(x-5)To find a, we need one more point through which the graph passes. Examining the diagram, we see that it's vertex is at (3.5,- 2). By substituting this point into the equation, we can solve for a.
c If a second degree equation can be factored, we are able to rewrite the coefficient of the x-term as a sum of two numbers, a and b, whose product equals the expression's constant.
y=x^2+(a+b)x+ab
⇕
y=(x+a)(x+b)
From the equation, we can identify what this sum and product must be.
cccccc
y&=&x^2 & + & 8 & -7pt x & + & 7
y&=&x^2 & + & (a+b) & -7pt x & + & abLet's list all of the ways that 7 can be factored and investigate which pair of numbers, a and b, that sum to 8. Notice that because both the constant and the product are positive, both a and b must be positive.
c|c|c
integer & ab & a+b [0.3em]
7 & 1(7) & 8
When a and b are 1 and 7, we have factored the equation.
y=x^2+8x+7 ⇔ y=(x+1)(x+7)
Having factored the equation, we can find the x-intercepts by setting y equal to 0 and solving for x.
The function has two x-intercepts at x=- 1 and x=- 7. Since this is a second degree equation, we can find the vertex by calculating the average of these intercepts.
average: - 1+(- 7)/2=- 4
When we know the vertex x-value we can find its y-value by substituting x=4 into the function.
The vertex is at (- 4,- 9). With this information, we can draw the graph.
d To find the maximum height and how long it takes to reach this height, we have to find the vertex. Notice that the function is a second degree equation. This means we can find the vertex if we find two points with the same y-coordinate such as its x-intercepts.
The function intercepts the x-axis at x=0 and x=8. By calculating the average of these, we can find the x-value of the vertex.
average: 0+8/2=4
The rocket reaches its maximum height after 4 seconds. By substituting this value into the function, we can determine the rocket's maximum height.
