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.

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Sure, let's analyze each part of the problem step by step. ### **Given:** $$ q(x) = g(x) + h(x) $$ Assume that both $$ g(x) $$ and $$ h(x) $$ are linear functions. Typically, a linear function can be expressed in the form: $$ g(x) = m_g x + b_g $$ $$ h(x) = m_h x + b_h $$ where: - $$ m_g $$ and $$ m_h $$ are the slopes of $$ g(x) $$ and $$ h(x) $$ respectively. - $$ b_g $$ and $$ b_h $$ are the $$ y $$-intercepts of $$ g(x) $$ and $$ h(x) $$ respectively. ### **a. Write the equation of $$ q(x) $$.** To find $$ q(x) $$, simply add $$ g(x) $$ and $$ h(x) $$: $$ q(x) = g(x) + h(x) = (m_g x + b_g) + (m_h x + b_h) $$ Combine like terms: $$ q(x) = (m_g + m_h)x + (b_g + b_h) $$ **Final Equation:** $$ q(x) = (m_g + m_h)x + (b_g + b_h) $$ --- ### **b. Relationship of the slope of $$ q(x) $$ to the slopes of $$ g(x) $$ and $$ h(x) $$.** **Slope of $$ q(x) $$:** $$ m_q = m_g + m_h $$ **Analysis:** 1. **Increasing or Decreasing:** - **If $$ m_q > 0 $$:** $$ q(x) $$ is increasing. - **If $$ m_q < 0 $$:** $$ q(x) $$ is decreasing. - **If $$ m_q = 0 $$:** $$ q(x) $$ is constant (horizontal line). 2. **Comparison of Slopes:** - **Increasing Faster:** If $$ m_g > m_h $$, then $$ g(x) $$ contributes more to the increase of $$ q(x) $$ than $$ h(x) $$. - **Decreasing Faster:** If either $$ m_g $$ or $$ m_h $$ is negative, the one with the larger absolute value affects the slope of $$ q(x) $$ more significantly. - **Same Rate:** If $$ m_g = m_h $$, both functions contribute equally to the slope of $$ q(x) $$. **Summary:** - The slope of $$ q(x) $$ is the sum of the slopes of $$ g(x) $$ and $$ h(x) $$. - $$ q(x) $$ increases or decreases based on whether $$ m_g + m_h $$ is positive or negative. - The function with the larger magnitude slope ($$ |m_g| $$ vs. $$ |m_h| $$) increases or decreases $$ q(x) $$ more rapidly. --- ### **c. Relationship of the $$ y $$-intercept of $$ q(x) $$ to the $$ y $$-intercepts of $$ g(x) $$ and $$ h(x) $$.** **$$ y $$-Intercept of $$ q(x) $$:** $$ b_q = b_g + b_h $$ **Explanation:** - The $$ y $$-intercept of $$ q(x) $$ is simply the sum of the $$ y $$-intercepts of $$ g(x) $$ and $$ h(x) $$. - This means that when $$ x = 0 $$, $$ q(0) = g(0) + h(0) = b_g + b_h $$. --- ### **d. Is $$ q(x) $$ the equation of a line? Justify your answer.** **Yes, $$ q(x) $$ is the equation of a line.** **Justification:** - Both $$ g(x) $$ and $$ h(x) $$ are linear functions, meaning their graphs are straight lines. - Adding two linear functions results in another linear function, as shown in part **a**: $$ q(x) = (m_g + m_h)x + (b_g + b_h) $$ - The form $$ q(x) = mx + b $$ is the standard equation of a straight line. - Therefore, $$ q(x) $$ represents a straight line. a. The equation of $$ q(x) $$ is $$ q(x) = (m_g + m_h)x + (b_g + b_h) $$. b. The slope of $$ q(x) $$ is the sum of the slopes of $$ g(x) $$ and $$ h(x) $$. If $$ m_g + m_h > 0 $$, $$ q(x) $$ is increasing; if $$ m_g + m_h < 0 $$, it is decreasing. The function with the larger absolute slope increases or decreases $$ q(x) $$ more rapidly. c. The $$ y $$-intercept of $$ q(x) $$ is the sum of the $$ y $$-intercepts of $$ g(x) $$ and $$ h(x) $$, i.e., $$ b_q = b_g + b_h $$. d. Yes, $$ q(x) $$ is the equation of a line because it is in the form $$ q(x) = mx + b $$, where $$ m = m_g + m_h $$ and $$ b = b_g + b_h $$.

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