Question

which of the following functions is an appropriate model for the data shown on the graph? linear, $f(x)=mx + b$ quadratic, $f(x)=ax^{2}+bx + c,a>0$ quadratic, $f(x)=ax^{2}+bx + c,a<0$

Explanation:

Step1: Analyze the shape of the data points

A linear function \( f(x)=mx + b \) has a straight - line graph. The given data points do not form a straight line.

Step2: Analyze the concavity for quadratic functions

For a quadratic function \( f(x)=ax^{2}+bx + c \), if \( a>0 \), the parabola opens upwards. If \( a < 0 \), the parabola opens downwards.
Looking at the general trend of the data points (assuming a parabolic - like shape), we can observe that the data points first decrease and then increase. A quadratic function with \( a>0 \) (parabola opening upwards) can model such a trend. A quadratic function with \( a < 0 \) (parabola opening downwards) would first increase and then decrease.

Answer:

B. quadratic, \( f(x)=ax^{2}+bx + c,a > 0 \)