i dont understand how to do it it is complicated and i need help

Question

Answer 1

A) The graph is a quadratic, not an exponential. Quadratics are U shaped parabolas such as y = ax^2 + bx + c. exponentials have the variable as an exponent such as y =ab^x. Quadratics have the variable as the base with the highest degreed term having an exponent of 2.

B) the a value is positive because the parabola is upward opening. If a were negative, it would be downward opening.

C) the c value is negative. It's the y intercept which is -3, it's the y coordinate of the point where the graph crosses the y axis

D) the vertex is the minimum point (1,-4) plug that into the general standard form of the equation for a parabola in vertex intercept form

y=a(x-h)^2 + k where (h,k) = the vertex = (1,-4)

y =a(x-1)^2 -4. Now plug in the y intercept (0,-3) to solve for a

-3 = a(0-1)^2 -4 = a-4

a =-3+4 = 1

y =(x-1)^2 -4 or

y = x^2 -2x+1 -4 =

y = x^2 -2x-3

4 A) the first two points in the chart are (-3,4.5) and (-2,2) the rate of change is the change in y coordinates divided by the change in x coordinates: (4.5-2)/(-3 - -2) = 2.5/(-3+2) = 2.5/-1 = -2.5. That's also the slope of a line through those two points.

4 B) rate of change between the 2nd and 3rd points (-2,2) and (0,0) is (2-0)/(-2-0) = 2/-2 = -1.

the chart has points on a parabola with vertex at the origin. points are symmetric about the y axis. 4th and 5th points are symmetric to the 1st and 2nd points. rate of change at a point is the slope of a tangent line through that point. The rate of change at the origin is zero = the slope of a tangent line through the parabola at the origin, (0,0) You can see the slope getting closer to zero, as you go from the 1st 2 points to the next 2 points, -2.5 to -1. keep going and it will be +1 for 3rd to 4th points and +2.5 for 4th & 5th points.

the parabola going through the 5 points is upward opening with vertex (0,0) and y intercept (0,0) y=a(x-0)^2 + 0 = ax^2

plug in any of the other 4 points to solve for a, such as (2,2)

y=ax^2

2=a(2)^2 = 4a

2=4a

a =2/4 = 1/2

the parabola through the points in the chart is y=(1/2)x^2 or 2y=x^2

4 A) and 4 B) are asking for the rate of change over an interval. with the parabola equation, solved for y, you can find the rate of change at a point (an interval of zero), an instantaneous rate of change, the derivative or y' = x

at (-3,4.5) the instantaneous rate of change is y'=x =-3

at (-2,2) the instantaneous rate of change is y'=x =-2

the rate of change over the interval between those two points is -2.5 or

the average of the two instantaneous rates of change: (-3-2)/2 =-5/2 = -2.5

same with the 2nd & 3rd points

at (-2,2) the instantaneous rate of change =-2

at (0,0) it's = 0

average of -2 and 0 is -1 which is the average rate of change over the

interval between those two points