Prior to this point, students have described characteristics of graphs, made sense of points on the graphs, and interpreted them in terms of a situation. In this lesson, students develop this work more formally, while continuing to use the idea of function as the focusing lens.
Students use mathematical terms such as intercept, maximum, and minimum in their graphical analyses, and relate features of graphs to features of the functions represented. For instance, they look at an interval in which a graph shows a positive slope and interpret that to mean an interval where the function’s values are increasing. Students also use statements in function notation, such as \(h(0)\) and \(h(t)=0\), to talk about key features of a graph.
By now, students are familiar with the idea of intercepts. Note that in these materials, the terms horizontal intercept and vertical intercept are used to refer to intercepts more generally, especially when a function is defined using variables other than \(x\) and \(y\). If needed, clarify these terms for students who may be accustomed only to using \(x\)-intercept and \(y\)-intercept.
As students look for connections across representations of functions and relate them to quantities in situations, they practice making sense of problems (MP1) and reasoning quantitatively and abstractly (MP2). Using mathematical terms and notation to describe features of graphs and features of functions calls for attention to precision (MP6).
- Analyze connections between statements that use function notation and features of graphs and describe (orally and in writing) these connections.
- Interpret key features of a graph—the intercepts, maximums, minimums, and the intervals when the function is increasing or decreasing—in terms of a situation.
- Understand and be able to use the terms “horizontal intercept,” “vertical intercept,” “maximum,” and “minimum” when talking about graphs of functions.
Let’s use graphs of functions to learn about situations.
- I can identify important features of graphs of functions and explain what they mean in the situations represented.
- I understand and can use the terms “horizontal intercept,” “vertical intercept,” “maximum,” and “minimum” when talking about functions and their graphs.
A function is decreasing if its outputs get smaller as the inputs get larger, resulting in a downward sloping graph as you move from left to right.
A function can also be decreasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is decreasing for \(x \ge 0\) because the graph slopes downward to the right of the vertical axis.
The horizontal intercept of a graph is the point where the graph crosses the horizontal axis. If the axis is labeled with the variable \(x\), the horizontal intercept is also called the \(x\)-intercept. The horizontal intercept of the graph of \(2x + 4y = 12\) is \((6,0)\).
The term is sometimes used to refer only to the \(x\)-coordinate of the point where the graph crosses the horizontal axis.
A function is increasing if its outputs get larger as the inputs get larger, resulting in an upward sloping graph as you move from left to right.
A function can also be increasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is increasing for \(x \le 0\) because the graph slopes upward to the left of the vertical axis.
A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the graph of the function is the corresponding highest point on the graph.
A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the graph of the function is the corresponding lowest point on the graph.
The vertical intercept of a graph is the point where the graph crosses the vertical axis. If the axis is labeled with the variable \(y\), the vertical intercept is also called the \(y\)-intercept.
Also, the term is sometimes used to mean just the \(y\)-coordinate of the point where the graph crosses the vertical axis. The vertical intercept of the graph of \(y = 3x - 5\) is \((0,\text-5)\), or just -5.
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