Earl runs 50 meters in 20 seconds. How many meters does Earl run per second?
2 answers:
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Let x refers to the Month numbers
And y refers to Number of births
There is 5 steps to find the quadratic function that models the data.
First step ⇒ <span>Plot <span>the data pairs (<span>x<span>, <span>y<span>) in a coordinate plane.
which are the blue circles in the attached figure
Second step ⇒ </span></span></span></span></span></span><span>Draw <span>the parabola you think best fits the data.
Which is the red function in the attached figure.
</span></span><span><span>Third step ⇒</span> Estimate <span>the coordinates of three points on the
<span> parabola, which are (2, 108), (4, 68), and (6, 75).</span>
</span></span><span><span>which are the black squares in the attached figure</span>
Fourth step ⇒ </span><span>Substitute <span>the coordinates of the points into the<span> model <span>
y <span>= <span>ax<span>2 <span>+ <span>bx <span>+ <span>c
<span> to obtain a system of three linear equations.</span>
4a + 2 b + c = 108 ---------(1)
</span></span></span></span></span></span></span></span></span></span></span><span><span>16a + 4 b + c = 68 ---------(2)</span>
36a + 6 b + c = 75 ---------(3)
Fifth step ⇒ </span><span>Solve <span>the linear system.
By using the calculator
a = 5.875 , b = -55.25 , c = 195
So, the</span></span><span> quadratic model for the data is ⇒ y = 5.875 x²- 55.25 x + 195
============================================
T</span><span>o predict the number of births for month 8
substitute with x = 8 ⇒ y = </span><span>5.875 * 8²- 55.25 * 8 + 195 = 129</span>
And y refers to Number of births
There is 5 steps to find the quadratic function that models the data.
First step ⇒ <span>Plot <span>the data pairs (<span>x<span>, <span>y<span>) in a coordinate plane.
which are the blue circles in the attached figure
Second step ⇒ </span></span></span></span></span></span><span>Draw <span>the parabola you think best fits the data.
Which is the red function in the attached figure.
</span></span><span><span>Third step ⇒</span> Estimate <span>the coordinates of three points on the
<span> parabola, which are (2, 108), (4, 68), and (6, 75).</span>
</span></span><span><span>which are the black squares in the attached figure</span>
Fourth step ⇒ </span><span>Substitute <span>the coordinates of the points into the<span> model <span>
y <span>= <span>ax<span>2 <span>+ <span>bx <span>+ <span>c
<span> to obtain a system of three linear equations.</span>
4a + 2 b + c = 108 ---------(1)
</span></span></span></span></span></span></span></span></span></span></span><span><span>16a + 4 b + c = 68 ---------(2)</span>
36a + 6 b + c = 75 ---------(3)
Fifth step ⇒ </span><span>Solve <span>the linear system.
By using the calculator
a = 5.875 , b = -55.25 , c = 195
So, the</span></span><span> quadratic model for the data is ⇒ y = 5.875 x²- 55.25 x + 195
============================================
T</span><span>o predict the number of births for month 8
substitute with x = 8 ⇒ y = </span><span>5.875 * 8²- 55.25 * 8 + 195 = 129</span>
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_______________
• Function: f(x) = 3x + 12.
A. Finding the inverse of f.
The composition of f with its inverse results in the identity function:
(f o g)(x) = x
f[ g(x) ] = x
3 · g(x) + 12 = x
3 · g(x) = x – 12
x – 12
g(x) = ⸺⸺
3
x
g(x) = ⸺ – 4 <——— this is the inverse of f.
3
________
B. Verifying that the composition of f and g gives us the identity function:
•
![\mathsf{=f\big[g(x)\big]}\\\\\\ \mathsf{=3\cdot \left(\dfrac{x}{3}-4\right)+12}\\\\\\ \mathsf{=\diagup\hspace{-7}3\cdot \dfrac{x}{\diagup\hspace{-7}3}-3\cdot 4+12}\\\\\\ \mathsf{=x-12+12}\\\\ \mathsf{=x\qquad\quad\checkmark}](https://tex.z-dn.net/?f=%5Cmathsf%7B%3Df%5Cbig%5Bg%28x%29%5Cbig%5D%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%3D3%5Ccdot%20%5Cleft%28%5Cdfrac%7Bx%7D%7B3%7D-4%5Cright%29%2B12%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdiagup%5Chspace%7B-7%7D3%5Ccdot%20%5Cdfrac%7Bx%7D%7B%5Cdiagup%5Chspace%7B-7%7D3%7D-3%5Ccdot%204%2B12%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3Dx-12%2B12%7D%5C%5C%5C%5C%0A%5Cmathsf%7B%3Dx%5Cqquad%5Cquad%5Ccheckmark%7D)
and also
•
![\mathsf{=g\big[f(x)\big]}\\\\\\ \mathsf{=\dfrac{f(x)}{3}-4}\\\\\\ \mathsf{=\dfrac{3x+12}{3}-4}\\\\\\ \mathsf{=\dfrac{\diagup\hspace{-7}3\cdot (x+4)}{\diagup\hspace{-7}3}-4}\\\\\\ \mathsf{=x+4-4}\\\\ \mathsf{=x\qquad\quad\checkmark}](https://tex.z-dn.net/?f=%5Cmathsf%7B%3Dg%5Cbig%5Bf%28x%29%5Cbig%5D%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%3D%5Cdfrac%7Bf%28x%29%7D%7B3%7D-4%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%3D%5Cdfrac%7B3x%2B12%7D%7B3%7D-4%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdfrac%7B%5Cdiagup%5Chspace%7B-7%7D3%5Ccdot%20%28x%2B4%29%7D%7B%5Cdiagup%5Chspace%7B-7%7D3%7D-4%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3Dx%2B4-4%7D%5C%5C%5C%5C%0A%5Cmathsf%7B%3Dx%5Cqquad%5Cquad%5Ccheckmark%7D)
________
C. Since f and g are inverse, then
f(g(– 2))
= (f o g)(– 2)
= – 2 <span>✔
</span>
• Call h the compositon of f and g. So,
h(x) = (f o g)(x)
h(x) = x
As you can see above, there is no restriction for h. Therefore, the domain of h is R (all real numbers).
I hope this helps. =)
_______________
• Function: f(x) = 3x + 12.
A. Finding the inverse of f.
The composition of f with its inverse results in the identity function:
(f o g)(x) = x
f[ g(x) ] = x
3 · g(x) + 12 = x
3 · g(x) = x – 12
x – 12
g(x) = ⸺⸺
3
x
g(x) = ⸺ – 4 <——— this is the inverse of f.
3
________
B. Verifying that the composition of f and g gives us the identity function:
•
and also
•
________
C. Since f and g are inverse, then
f(g(– 2))
= (f o g)(– 2)
= – 2 <span>✔
</span>
• Call h the compositon of f and g. So,
h(x) = (f o g)(x)
h(x) = x
As you can see above, there is no restriction for h. Therefore, the domain of h is R (all real numbers).
I hope this helps. =)