Question

Make a scatter plot of the data. Then tell whether a linear, exponential, or quadratic model fits the data. (Review 9.81) $$(-1,16),(0,4),(1,-2),(2,-2),(3,4),(5,34)$$

Short answer

The scatter plot of the data forms a U-shape, which indicates that a quadratic model probably fits the data best.

Step-by-step solution

Plot the given data points

Start by drawing a scatter plot and plot each of the given points on it: \((-1,16), (0,4), (1,-2), (2,-2), (3,4), (5,34)\). This process will visually represent all the coordinates provided.

Analyze the scatter plot

Once the scatter plot has been drawn, look for patterns in the distribution of points. If the points form a straight line, then they could be best described by a linear model. If the points increase or decrease rapidly, then an exponential model might be best. If the points form a parabola (U-shape or inverted U-shape), then a quadratic model could fit best.

Determine the best fitting model

After closely inspecting the scatter plot, it should be clear that the points form a kind of U-shape. This means that the data is likely best represented by a quadratic model.

Key concepts

Linear Model

In the context of scatter plots and data analysis, a linear model represents a trend that can be described by a straight line. This model is typically characterized by a constant rate of change or slope. It suggests that for every one unit increase in the independent variable (usually represented on the x-axis), there is a consistent increase or decrease in the dependent variable (y-axis).

For example, if a set of data points on a scatter plot form a line that rises or falls at a steady rate, then a linear equation of the form \( y = mx + b \) would likely be a good fit, where \( m \) represents the slope and \( b \) the y-intercept. In real-world scenarios, linear models are often used to predict outcomes such as sales over time, temperature changes, or distance traveled at a constant speed.

Considering the given exercise with data points \((-1,16), (0,4), (1,-2), (2,-2), (3,4), (5,34)\), if these points had formed a straight line when plotted, a linear model would have been the appropriate choice. However, as they do not, we must explore other models.

Exponential Model

An exponential model is characterized by rapid increases or decreases in the dependent variable in relation to the independent variable and is often used when the rate of change is not constant but rather grows or shrinks at a proportionate rate.

An exponential equation is typically of the form \( y = ab^x \) where \( a \) is a constant that represents the y-intercept, and \( b \) is the base of the exponent that determines the rate of growth or decay. The model is a powerful tool for describing phenomena such as population growth, radioactive decay, or interest compounding in finance.

For the dataset provided in the exercise, should it have displayed a consistent pattern of rapid growth or decay as we moved along the x-axis, an exponential model would have been a compelling choice. However, as the growth of our data points is not consistent or proportional, we find the exponential model to be an unsuitable representation of the given data.

Quadratic Model

The quadratic model is apt for datasets where the relationship between variables forms a parabolic shape, which can either open upwards or downwards. Represented by the general form \( y = ax^2 + bx + c \), a quadratic model includes a squared term which allows for the curvature that is indicative of acceleration or deceleration in the dependent variable.

This type of model is commonly applicable in situations involving gravity, such as the trajectory of a projectile, or economic scenarios, like the relationship between supply and demand leading to changes in pricing. The parameters \( a \), \( b \), and \( c \) in the quadratic equation determine the shape and position of the parabola.

Returning to our exercise, after plotting the given data points, we observed a U-shaped curved pattern. This suggested motion, either up or down, indicated the presence of acceleration or deceleration – a hallmark of quadratic relationships. Consequently, the quadratic model is the most suitable representation for our scatter plot, capturing the nuanced changes in our data set effectively.