The graph shows the amount of water in a 100 -gallon tank...

The graph shows the amount of water in a 100 -gallon tank after $x$ minutes have elapsed. (GRAPH CANNOT COPY) (A) Is water entering or leaving the tank? How much water is in the tank after 3 minutes? (B) Find the $x$ - and $y$ -intercepts. Interpret each intercept. (C) Find the slope-intercept form of the equation of the line. Interpret the slope. (D) Estimate the $x$ -coordinate of the point $(x, 50)$ that lies on the line.

The graph shows the amount of water in a 100 -gallon tank after $x$ minutes have elapsed.
(GRAPH CANNOT COPY)
(A) Is water entering or leaving the tank? How much water is in the tank after 3 minutes?
(B) Find the $x$ - and $y$ -intercepts. Interpret each intercept.
(C) Find the slope-intercept form of the equation of the line. Interpret the slope.
(D) Estimate the $x$ -coordinate of the point $(x, 50)$ that lies on the line.

Key Concepts

Linear Functions

A linear function models a constant rate of change between two variables, represented graphically as a straight line. It is expressed in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Understanding linear functions is crucial for analyzing relationships where one quantity changes at a fixed rate relative to another.

Slope

The slope of a line represents the rate of change between the independent and dependent variables. In real-world contexts, the slope indicates how fast or slow the change occurs, such as water entering or leaving a tank per minute. It quantifies the relationship between time and the amount of water, with a positive slope indicating an increase and a negative slope indicating a decrease.

Intercepts

Intercepts are the points where a graph crosses the axes. The y-intercept is where the line intersects the vertical axis, representing the initial value of the dependent variable when the independent variable is zero. The x-intercept is where the graph crosses the horizontal axis, representing the point in time when the dependent variable has reached zero. These intercepts provide important contextual information about the starting conditions and the time it takes for the variable to reach a specific value.

Rate of Change

Rate of change is a measure of how one quantity changes relative to a change in another quantity. It is a fundamental concept in algebra and calculus used to describe the speed at which something happens, such as the flow rate of water in a tank. In the context of a graph, the rate of change is visually interpreted as the slope of the line, allowing for both quantitative calculations and qualitative understanding of trends.

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