First, we should analyze what the question gives us. The question states that there are x-intercepts (roots) at 2 and 7. It also states that there is a MAXIMUM value of 25.
Now, we convert that information into math terms:
x = 2 --> This can be rearranged to be (x-2). Likewise, (x-7) is also a root. Put them together and we have
(x-2) (x-7) = x2 - 9x + 14. Since we know that there is a maximum value, the function opens downwards meaning that we should multiply the function by -1 (Think downwards function is a sad face and being sad is negative). This gives us the new function of -x2 + 9x -14.
What should be done now is obtain the current maximum value. To do that, we must find the x-value at which the highest point is. This can be done by finding the distance between the two roots, 7 - 2 = 5, halving that distance due to symmetry of the curve, 5 / 2 = 2.5, then adding the distance between the origin and the leftmost root, 2 + 2.5 = 4.5. This methodology applies only to quadratic functions that do not open/close over the origin, but both cases can be solved due to geometry and reasoning.
Knowing both the formula and the x-value, we can put them together and solve for the current maximum for the y-value:
The maximum value is 6.25, but we want it to be 25, so we must find how much larger 25 is than 6.25 (I am letting "t" be the symbol for scale factor in this case as I do not know what your teacher uses):
6.25 x t = 25 --> t = 25 / 6.25 --> t = 4
25 is 4 times larger than 6.25, so we must make the function 4 times larger by multiplying it by 4:
y = 4(-x2 + 9x - 56) = -4x2 + 36x - 56
That final function would then be your answer. I explained my entire logical processing for this problem, so you can probably skim past the parts you understand and get right to where you might be having troubles.
I encourage other tutors to also explain their reasoning because students should be encouraged to understand concepts rather than memorize and repeat as that is how errors occur, so although showing the work or simply writing the answer is technically true, it doesn't do anything other than show off that you can do high school math.
My qualifications are that I am a Mechanical Engineering student at the University of Calgary and have been working in fields of education for over 4 years.
5 years ago
Answered By Jackie C
The other way
y=K(x-2)(x-7), K=y/[(x-2)*(x-7)]
To apply max y at symmetry condition with x=0.5*(2+7)=4.5, then
5 years ago
Answered By Jackie C
y=ax^2+bx+c
x1=2, y=0, then a*2^2+b*2+c=4a+2b+c, so
4a+2b+c=0.......(1)
x2=7, y=0, then a*7^2+b*7+c=49a+7b+c, so
49a+7b+c=0.....(2)
symmetry condition with max y=25, i.e., x3=0.5*(x1+x2)=4.5, then a*4.5^2+b*4.5+c=20.25a+2.5b+c, so
20.25a+2.5b+c=25......(3)
From (1), (2), and (3), you get
a=-4
b=36
c=-56
y=-4x^2+36x-56
5 years ago
Answered By Jackie C
Equation (3) should be
20.25a+4.5b+c=25
Sorry for the typo
5 years ago
Answered By Majid B
Use the vertex form of a quadratic function:
$y=a\left(x-p\right)^2+q$y=a(x−p)2+q
Because maximum value of function is 25:
$q=25$q=25
Replace coordinates of x-interception points in the equation: (2,0),(7,0)
$0=a\left(2-p\right)^2+25$0=a(2−p)2+25
$0=a\left(7-p\right)^2+25$0=a(7−p)2+25
Subtract equations:
$0-0=\left(a\left(2-p\right)^2+25\right)-\left(a\left(7-p\right)^2+25\right)$0−0=(a(2−p)2+25)−(a(7−p)2+25)
$0=a\left(2-p\right)^2-a\left(7-p\right)^2$0=a(2−p)2−a(7−p)2
$a\left(2-p\right)^2=a\left(7-p\right)^2$a(2−p)2=a(7−p)2
$\left(2-p\right)^2=\left(7-p\right)^2$(2−p)2=(7−p)2
$2-p=\pm\left(7-p\right)$2−p=±(7−p)
$2-p_1=7-p_1$2−p1=7−p1
$2=7$2=7 (Not Acceptable)
$2-p_2=-\left(7-p_2\right)$2−p2=−(7−p2)
$2+7=2p_2$2+7=2p2
$p_2=4.5$p2=4.5
Replace p = 4.5 in the first equation:
$0=a\left(2-4.5\right)^2+25$0=a(2−4.5)2+25
$-25=a\left(-2.5\right)^2$−25=a(−2.5)2
$a=\frac{-25}{6.25}=-4$a=−256.25 =−4
Final answer:
$a=-4$a=−4
$p=4.5$p=4.5
$q=25$q=25
$y=-4\left(x-4.5\right)^2+25$y=−4(x−4.5)2+25
5 years ago
Answered By Daunte S
First, we should analyze what the question gives us. The question states that there are x-intercepts (roots) at 2 and 7. It also states that there is a MAXIMUM value of 25.
Now, we convert that information into math terms:
x = 2 --> This can be rearranged to be (x-2). Likewise, (x-7) is also a root. Put them together and we have
(x-2) (x-7) = x2 - 9x + 14. Since we know that there is a maximum value, the function opens downwards meaning that we should multiply the function by -1 (Think downwards function is a sad face and being sad is negative). This gives us the new function of -x2 + 9x -14.
What should be done now is obtain the current maximum value. To do that, we must find the x-value at which the highest point is. This can be done by finding the distance between the two roots, 7 - 2 = 5, halving that distance due to symmetry of the curve, 5 / 2 = 2.5, then adding the distance between the origin and the leftmost root, 2 + 2.5 = 4.5. This methodology applies only to quadratic functions that do not open/close over the origin, but both cases can be solved due to geometry and reasoning.
Knowing both the formula and the x-value, we can put them together and solve for the current maximum for the y-value:
y(4.5) = -x2 + 9x -14 = -(4.5)2 + 9(4.5) - 14 = 6.25
The maximum value is 6.25, but we want it to be 25, so we must find how much larger 25 is than 6.25 (I am letting "t" be the symbol for scale factor in this case as I do not know what your teacher uses):
6.25 x t = 25 --> t = 25 / 6.25 --> t = 4
25 is 4 times larger than 6.25, so we must make the function 4 times larger by multiplying it by 4:
y = 4(-x2 + 9x - 56) = -4x2 + 36x - 56
That final function would then be your answer. I explained my entire logical processing for this problem, so you can probably skim past the parts you understand and get right to where you might be having troubles.
I encourage other tutors to also explain their reasoning because students should be encouraged to understand concepts rather than memorize and repeat as that is how errors occur, so although showing the work or simply writing the answer is technically true, it doesn't do anything other than show off that you can do high school math.
My qualifications are that I am a Mechanical Engineering student at the University of Calgary and have been working in fields of education for over 4 years.
5 years ago
Answered By Jackie C
The other way
y=K(x-2)(x-7), K=y/[(x-2)*(x-7)]
To apply max y at symmetry condition with x=0.5*(2+7)=4.5, then
K=25/[(4.5-2)*(4.5-7)]=-4
y=(-4)(x-2)(x-7)=-4x^2+36x-56