Question

describing a model for a graph
plant growth
the graph shows the recorded heights of a plant in a soil experiment. which statements describe the plant height function? check all that apply.
the function is linear.
the function is quadratic.
the height increases at a constant rate of change.
the ratio of height and number of weeks is constant.
there is no relationship between height and number of weeks.

Explanation:

Step1: Check linearity

A linear function has a constant rate of change and its graph is a straight - line. By observing the points on the graph, we can calculate the rate of change between consecutive points.

Step2: Calculate rate of change

Let's take two points \((t_1,h_1)\) and \((t_2,h_2)\). For example, taking \((0,2)\) and \((2, 2.5)\), the rate of change \(m=\frac{h_2 - h_1}{t_2 - t_1}=\frac{2.5 - 2}{2-0}=\frac{0.5}{2}=0.25\). Taking another pair \((2,2.5)\) and \((4,3)\), \(m=\frac{3 - 2.5}{4 - 2}=\frac{0.5}{2}=0.25\). The rate of change is constant, so the function is linear.

Step3: Analyze quadratic nature

A quadratic function has a non - constant second - derivative and its graph is a parabola. Since the rate of change is constant, it is not quadratic.

Step4: Check ratio constancy

The ratio of height \(h\) and number of weeks \(t\) is not constant. For \(t = 0,h = 2\), the ratio is undefined. For \(t=2,h = 2.5\), the ratio is \(\frac{2.5}{2}=1.25\), for \(t = 4,h=3\), the ratio is \(\frac{3}{4}=0.75\).

Step5: Check relationship existence

Since there is a constant rate of change, there is a relationship between height and number of weeks.

Answer:

The function is linear.
The height increases at a constant rate of change.