I want to confirm my work. Thanks in advance for the help.
#1. Write the quadratic function in general form given that the function has an x-intercept of 6 and 1, and a y-intercept of 9.
x= 6, 1
y= 9
(0,9)
y=a(x-r1)(x-r2)
9=a(0-6)(0-1)
9=a(-6)(-1)
9=a(6)
a=3/2
y= 3/2(x-6)(x-1)
= 3/2(x^2-x-6x+6)
= 3/2(x^2-7x+6)
= (3/2x^2) - (21/2x) + 18/2
y= (3/2x^2) - (21/2x) + 9
#2. A small nerf gun was shot into the air by the function h(t) = 3(t-1)^2+2, where h is height and t is time.
How many seconds after it was fired did the nerf bullet hit the floor?
h(t) is 0 due to it landing, therefore there is no height.
0 =3(x-1)^2-4
=3x^2-6x-1
0 = -b ± √(b^2 - 4ac)/2a
= +6 ± √[-6^2 - 4(3)(-1)]/2(3)
= 2.15, -0.15
No negative time, therefore answer is 2.15 seconds.
#3. A cafe sells a cup of coffee for $3.00. The cafe sells around 800 cups per month. The poll shows that for every $0.50 increase in price, the cafe will sell 20 less cups of coffee. Find the price and number of cups sold that will maximise revenue.
R=(3.00+0.5x)(800-20x)
= 2400-60x+400x -10x^2
= -10x^2 + 340x +2400
= -10(x^2-34x)+2400
= -10(x-17)^2 +2400+2890
= -10(x-17)^2 +5290
Cups: 800-20(17)=460
Price: 3.00-0.5(2)=$11.5
#1 ok
#2 h(t) = 3(t-1)^2+2
is apparently a typo, but your solution is correct as written
#3 ok
if you ignore 3.00-0.5(2)=$11.5
good work
To confirm your work, let's go through each problem:
#1:
To write the quadratic function in general form with the given x-intercepts and y-intercept, you correctly used the vertex form of a quadratic equation, which is y = a(x - r1)(x - r2). You plugged in the x-intercepts (6 and 1) and the y-intercept (9) into the equation and solved for the constant "a". You correctly determined that "a" is 3/2. Then, you expanded the equation and simplified it to obtain y = (3/2x^2) - (21/2x) + 9. Your final answer is correct.
#2:
To find out how many seconds after it was fired did the nerf bullet hit the floor, you correctly set the equation for height (h(t)) equal to zero and solved for t. You correctly expanded the equation, simplified it, and used the quadratic formula to find the roots. You correctly determined that the solution is t = 2.15 seconds. Your answer is correct.
#3:
To find the price and number of cups sold that will maximize revenue, you correctly set up the revenue equation as R = (3.00 + 0.5x)(800 - 20x), where x is the price increase in dollars. You then expanded the equation and simplified it to obtain R = -10(x - 17)^2 + 5290. You determined that to maximize revenue, the price increase should be $2 and the number of cups sold would be 460. However, the final price you mentioned, $11.5, seems to be a typo or miscalculation. The correct final price should be $3.00 + (0.5 * 2) = $4.00. Therefore, the correct answer is that the price and number of cups sold to maximize revenue are $4.00 and 460 cups, respectively.
Overall, your work is mostly correct. Just double-check the final price in problem #3 to ensure accuracy. Well done!