Modeling With Quadratic Functions Worksheet - Section 2.4 Page 3
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8Writing an Equation Using a Point and x-Intercepts
A meteorologist creates a parabola to predict the temperature tomorrow, where x
Temperature Forecast
is the number of hours after midnight and y is the temperature (in degrees Celsius).
y
a. Write a function f that models the temperature over time. What is the coldest
(0, 9.6)
10
temperature?
(4, 0)
(24, 0)
b. What is the average rate of change in temperature over the interval in which the
0
x
3
9
15
temperature is decreasing? increasing? Compare the average rates of change.
−10
SOLUTION
Hours after midnight
a. The x-intercepts are 4 and 24 and the parabola passes through (0, 9.6). Use the
x-intercepts and the point to solve for a in intercept form.
y = a(x − p)(x − q)
Intercept form
9.6 = a(0 − 4)(0 − 24)
Substitute for p, q, x, and y.
9.6 = 96a
Simplify.
0.1 = a
Divide each side by 96.
Because a = 0.1, p = 4, and q = 24, the temperature over time can be modeled
by f(x) = 0.1(x − 4)(x − 24), where 0 ≤ x ≤ 24. The coldest temperature is the
4 + 24
minimum value. So, fi nd f(x) when x =
= 14.
—
2
f (14) =
−
− 24)
Substitute 14 for x.
0.1(14
4)(14
REMEMBER
= −10
Simplify.
The average rate of
So, the coldest temperature is −10°C at 14 hours after midnight, or 2 p.m.
change of a function f
from x
to x
is the slope
b. The parabola opens up and the axis of symmetry is x = 14. So, the function is
1
2
of the line connecting
decreasing over the interval 0 < x < 14 and increasing over the interval 14 < x < 24.
(x
, f(x
)) and (x
, f(x
)):
1
1
2
2
Average rate of change
Average rate of change
f(x
) – f(x
)
over 0 < x < 14:
over 14 < x < 24:
2
1
.
——
x
– x
2
1
f(14) − f(0)
−10 − 9.6
f(24) − f(14)
0 − (−10)
=
= −1.4
=
= 1
—
—
——
—
14 − 0
24 − 14
14
10
y
(0, 9.6)
10
(24, 0)
0
3
15
x
−10
(14, −10)
∣
∣
∣
∣
−1.4
Because
>
1
, the average rate at which the temperature decreases
from midnight to 2 p.m. is greater than the average rate at which it increases
from 2 p.m. to midnight.
Monitoring Progress
Monitoring Progress
Help in English and Spanish at
3.
WHAT IF?
The y-intercept is 4.8. How does this change your answers in
parts (a) and (b)?
4.
Write an equation of the parabola that passes through the point (2, 5) and has
x-intercepts −2 and 4.
Section 2.4
Modeling with Quadratic Functions
77
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