Question

example 2 graph a power function by using a table
pressure for water to flow through a garden hose at a certain rate
in gallons per minute (gpm), it needs to have a specific pressure in
pounds per square inch (psi). through testing and measurement,
a company that produces garden hoses determines that the pressure
p given the flow rate f is defined by $p(f) = \frac{3}{2}f^2$. graph the function
p(f), and state the domain and range.

steps 1 and 2 find a and n. then state the domain and range.

for $p(f) = \frac{3}{2}f^2$, a = ____, and n = ____.

the domain is ____________ because a is positive and n is even,
the range is all real numbers ______.

steps 3–5 create a table of values and graph the ordered pairs.

f	$\frac{3}{2}f^2$	p(f)
-1	$\frac{3}{2}(-1)^2$	1.5
0	$\frac{3}{2}(0)^2$
1	$\frac{3}{2}(1)^2$
2	$\frac{3}{2}(2)^2$	6

(graph: pressure (psi) on y - axis, flow rate (gpm) on x - axis, with a parabola - like graph plotted)

Explanation:

Step1: Identify \( a \) and \( n \) in power function

A power function is in the form \( y = a x^n \). For \( P(F)=\frac{3}{2}F^2 \), comparing with \( y = a x^n \), we have \( a=\frac{3}{2} \) and \( n = 2 \).

Step2: Determine domain

The flow rate \( F \) (in gpm) can be any real number (since flow rate can be positive, negative in theory for the function, but in context, it's non - negative, but mathematically for the function \( P(F)=\frac{3}{2}F^2 \), \( F \) can take any real value). So the domain is all real numbers.

Step3: Determine range

Since \( a=\frac{3}{2}>0 \) and \( n = 2 \) (even), the function \( P(F)=\frac{3}{2}F^2 \) is a parabola opening upwards. The minimum value of \( F^2 \) is \( 0 \) (when \( F = 0 \)), so the minimum value of \( P(F) \) is \( \frac{3}{2}(0)^2=0 \). And as \( |F| \) increases, \( P(F) \) increases. So the range is all real numbers greater than or equal to \( 0 \).

Step4: Calculate \( P(F) \) for \( F=-2 \)

For \( F=-2 \), \( P(F)=\frac{3}{2}(-2)^2=\frac{3}{2}\times4 = 6 \).

Step5: Calculate \( P(F) \) for \( F = 0 \)

For \( F = 0 \), \( P(F)=\frac{3}{2}(0)^2=0 \).

Step6: Calculate \( P(F) \) for \( F = 1 \)

For \( F = 1 \), \( P(F)=\frac{3}{2}(1)^2=\frac{3}{2}=1.5 \).

Answer:

For \( P(F)=\frac{3}{2}F^2 \), \( a=\boldsymbol{\frac{3}{2}} \), and \( n=\boldsymbol{2} \).
The domain is \(\boldsymbol{\text{all real numbers}}\). Because \( a \) is positive and \( n \) is even, the range is all real numbers \(\boldsymbol{\text{greater than or equal to } 0}\).
For the table:

• When \( F=-2 \), \( P(F)=6 \)
• When \( F = 0 \), \( P(F)=0 \)
• When \( F = 1 \), \( P(F)=1.5 \)