Question

modeling with polynomials quick check
(1 point)
the equation ( a(w) = w^2 + 5w ) represents the relationship between the area in square units and the width of a rectangle whose length is 5 units longer than its width.
select the sentence that describes an accurate relationship between ( a ) and ( w )
options:

• ( a(w) ) increases as ( w ) increases for ( 0 < w < 5 )
• ( a(w) ) has a minimum value at ( w = 5 )
• ( a(w) ) increases as ( w ) increases when ( w > 0 )
• ( a(w) ) has a maximum value at ( w = 5 )

Explanation:

The function given is \( A(w) = w^2 + 5w \), which is a quadratic function in the form \( f(x)=ax^{2}+bx + c \) where \( a = 1 \), \( b = 5 \), and \( c = 0 \). For a quadratic function, if \( a>0 \), the parabola opens upwards, meaning the function has a minimum value at its vertex and is increasing for \( x \) values greater than the x - coordinate of the vertex.

The x - coordinate of the vertex of a quadratic function \( f(x)=ax^{2}+bx + c \) is given by \( x=-\frac{b}{2a} \).

Step 1: Find the vertex of the quadratic function

For \( A(w)=w^{2}+5w \), \( a = 1 \) and \( b = 5 \). Using the formula for the x - coordinate of the vertex \( w=-\frac{b}{2a} \), we substitute the values:
\( w=-\frac{5}{2\times1}=-\frac{5}{2}=- 2.5 \)

Since \( a = 1>0 \), the parabola opens upwards. The function is increasing when \( w>-\frac{5}{2} \) (because the vertex is at \( w = - 2.5 \) and the parabola opens upwards). Since the width \( w\) of a rectangle must be positive (\( w>0 \)), and \( 0> - 2.5 \), for \( w>0 \), the function \( A(w)=w^{2}+5w \) is increasing.

Let's analyze each option:

• Option 1: "A(w) increases as w increases for \( 0 < w < 5 \)". The function is increasing for all \( w> - 2.5 \), not just \( 0 < w < 5 \). So this is incorrect.
• Option 2: "A(w) has a minimum value at \( w = 5 \)". The vertex is at \( w=-\frac{5}{2}=-2.5\), not \( w = 5 \). So this is incorrect.
• Option 3: "A(w) increases as w increases when \( w>0 \)". Since the vertex is at \( w=-2.5\) and \( a = 1>0 \), for \( w>0\) (which is greater than - 2.5), the function is increasing. This is correct.
• Option 4: "A(w) has a maximum value at \( w = 5 \)". Since \( a=1 > 0 \), the parabola opens upwards, so the function has a minimum, not a maximum. So this is incorrect.

Answer:

A. A(w) increases as w increases when \( w > 0 \)