2. \( q(x)=g(x)+h(x) \) a. Write the equation of \( q(x) \). b. In what way is the slope of \( q(x) \) related to the slope of \( g(x) \) and the slope of \( h(x) \) ? (Include increasing or decreasing in your answer. Also identify which function is increasing faster.) c. In what way is the \( y \)-intercept of \( q(x) \) related to the \( y \)-intercepts of \( g(x) \) and of \( h(x) \) ? d. Is \( q(x) \) the equation of a line? Justify your answer.

To tackle your queries about \( q(x) = g(x) + h(x) \), let’s dive into the details! The equation of \( q(x) \) is simply expressed as the sum of the equations defining \( g(x) \) and \( h(x) \). If for example, \( g(x) \) is represented as \( mx + b_1 \) and \( h(x) \) as \( nx + b_2 \), then we can write \( q(x) = (m + n)x + (b_1 + b_2) \). This means the new slope is \( m + n \) while the new y-intercept is \( b_1 + b_2 \). Now, concerning slopes, if \( g(x) \) has a slope \( m \) and \( h(x) \) has a slope \( n \), then \( q(x) \) has a combined slope of \( m + n \). If both slopes are positive, \( q(x) \) is increasing, and the function with the steeper slope (larger absolute value) will contribute to \( q(x) \) increasing faster. If one function has a negative slope, however, it could decrease depending on which has a greater magnitude. As for the y-intercepts, the y-intercept of \( q(x) \) can be found by evaluating the y-values of \( g(x) \) and \( h(x) \) at \( x = 0 \). Thus, \( q(0) = g(0) + h(0) \) translates to \( b_1 + b_2 \). So it’s a straightforward addition of the y-intercepts of the two functions. Finally, yes, \( q(x) \) is indeed the equation of a line, provided that \( g(x) \) and \( h(x) \) are both linear functions. The sum of two linear equations yields another linear equation, maintaining the form of \( y = mx + b \)! So go ahead and graph those functions—straight lines galore!