The table presents a relationship between age (in years) and height (in centimeters). The task asks to determine the function that describes height as a function of age.

Step-by-step approach:

1. Data points from the table:

   • Age 6 → Height 119 cm
   • Age 8 → Height 132 cm
   • Age 10 → Height 143 cm
   • Age 12 → Height 155 cm
   • Age 14 → Height 167 cm

2. Assume the relationship is linear: Let the relationship between height $h$ and age $a$ be described by the linear function: $h(a) = m \cdot a + b$ where $m$ is the slope and $b$ is the y-intercept.

3. Calculate the slope $m$: The slope $m$ is the rate of change in height with respect to age, which can be calculated as: $m = \frac{h_2 - h_1}{a_2 - a_1}$ Using two points, say (6, 119) and (8, 132): $m = \frac{132 - 119}{8 - 6} = \frac{13}{2} = 6.5$ So, the slope $m = 6.5$.

4. Find the y-intercept $b$: Using the point (6, 119) and substituting into the linear equation: $119 = 6.5 \cdot 6 + b$ $119 = 39 + b$ $b = 119 - 39 = 80$

5. Final function: The function that describes height as a function of age is: $h(a) = 6.5 \cdot a + 80$

This equation predicts the height for a given age. Would you like further explanation or details on how to use this equation for predictions?

Here are 5 related questions to explore:

1. How would you verify if the function is a good fit for all data points?
2. Can this relationship still be valid for ages beyond the given data range?
3. What is the interpretation of the slope in this context?
4. How would the equation change if the relationship was quadratic instead of linear?
5. How can you calculate the age when the height is expected to be 180 cm?

Tip: Always double-check the function by substituting known data points into it.