We've got the functions f(x) = ax^2 + 8 and g(x) = 3x^2 - bx
a. For one value of the parameter a is the accompanying graph not an parabola. For which value of a is that the case and what for a function is it then?
b. For the other values of a is the graph of f(x) = ax^2 + 8 a parabola. Which coordinates does the top(vertex?) of each of those graphs have?
c. For which value of a does the graph of f(x) = ax^2 + 8 go through the points (-3, 5)?
d. For which values of a is the graph of f(x) = ax^2 + 8 a parabola opening downward that is more narrow than the ones that belong to g(x) = 3x^2 - bx?
e. The graph of g(x) = 3x^2 - bx cuts the horizontal axis in a point where the x-coordinate is equal to -2. Which value of parameter b belongs to this?
+6252 a) a=0 makes f(x) a constant function
b) \(\text{the "top" of } f(x) \text{ is }f(0)=0 \\\text{ the "top" of }g(x) \text{ is } f\left(\dfrac b 6\right) =-\dfrac{b^2}{12} \)
c) \(a(-3)^2 + 8 = 5\\ 9a = -3\\ a=-\dfrac 1 3\)
d) \(\text{the coefficient of the }x^2 \text{ term controls the narrowness of the parabola}\\ \text{the larger absolute value of the coefficient the narrower the parabola}\\ \text{a negative value produces a downward facing parabola, thus}\\ a < -3 \text{ will produce a downward facing parabola that is narrower than g(x)}\)
e) \(3(-2)^2 - b(-2) = 0\\ 12 +2 b = 0\\ b=-6\)
