Let's analyze and solve the questions from the image you provided. Here’s a breakdown for each problem:

Question 5:

Explain why the values of the exponential expression $3^x$ will eventually overtake the values of the quadratic expression $10x^2$.

Explanation:

• Exponential functions grow much faster than polynomial functions as $x$ increases.
• For smaller values of $x$, $3^x$ may be less than $10x^2$, since the quadratic $10x^2$ grows quickly for small values of $x$.
• However, as $x \to \infty$, the base of the exponential function ($3$) is repeatedly multiplied by itself, leading to much larger growth compared to the quadratic ($x^2$), which only grows by squaring $x$.
• Mathematically, the growth of an exponential function is proportional to its current value, making it faster than the polynomial, whose growth is proportional to a lower power of $x$.

Question 6:

A baseball travels $d$ meters $t$ seconds after being dropped from the top of a building. The distance traveled by the baseball can be modeled by $d = 5t^2$. Which graph represents this situation?

• The equation $d = 5t^2$ describes a quadratic relationship between distance $d$ and time $t$, meaning the graph should show a curve where $d$ increases more rapidly as $t$ increases.
• Comparing Graph A and Graph B:
  • Graph A shows a linear relationship, which is incorrect because $d = 5t^2$ is quadratic, not linear.
  • Graph B shows a curve, consistent with the quadratic nature of $d = 5t^2$.
• Correct Answer: Graph B represents this situation.

Why:

• For each second that passes, the increase in distance becomes larger due to the squared term ($t^2$).

Question 7:

Consider a function $q$ defined by $q(x) = x^2$. Explain why negative values are not included in the range of $q$.

Explanation:

• The range of a function refers to the set of all possible output values.
• For $q(x) = x^2$:
  • Squaring any real number $x$ (positive, negative, or zero) results in a non-negative value.
  • For example:
    • $(-2)^2 = 4$
    • $0^2 = 0$
    • $2^2 = 4$
  • There are no values of $x$ that produce a negative result when squared.
• Therefore, the range of $q(x) = x^2$ is $[0, \infty)$, and negative values cannot be included.

Follow-Up Questions:

1. What happens when the base of an exponential function (like $3^x$) is less than 1? How does this affect its growth?
2. Can you identify the vertex of $d = 5t^2$ and explain its physical meaning in the context of the problem?
3. How does the function $q(x) = x^2$ change when shifted vertically (e.g., $q(x) = x^2 + 3$)? What happens to its range?
4. Why does the rate of change in $3^x$ increase as $x$ increases, but the rate of change in $10x^2$ stays constant for large values of $x$?
5. If $d(t) = 5t^2$, how would the graph change if the coefficient of $t^2$ were different (e.g., $d(t) = 2t^2$)?

Tip:

Always compare the growth rates of functions by calculating or estimating their derivatives—this provides insight into how quickly each function changes relative to the other. For example, the derivative of $3^x$ is proportional to $3^x$, but the derivative of $10x^2$ is linear.