In this section, students are introduced to patterns that grow quadratically. They compare these new patterns to linear and exponential patterns from earlier units and recognize that a new type of relationship is needed. They work with rectangular areas that have a given perimeter and visual patterns that grow quadratically with each step and use tables, graphs, and expressions to describe the relationships.

Throughout the section there are opportunities to use technology to solve problems. If possible, make these tools available so that students can focus on the pattern itself rather than calculations or visualization.

Three steps of a growing pattern. Step 1: two squares, one atop the other. Step 2: two squares on row 1, two squares on row 2 and one square on the left on row 3. Step 3: three squares on row 1, three squares on Row 2, 3 squares on Row 3 and 1 square on the left on row 4.

In this section, students examine quadratic relationships in the context of functions. They compare quadratic-function growth to that of linear and exponential functions. They notice that quadratic functions with a positive quadratic coefficient grow faster than linear functions and grow slower than exponential functions that have a growth factor greater than 1.

Then, students explore quadratic functions in contexts such as objects in free-fall, projectile motion, and revenue curves. They use the structure of the functions to determine the zeros, intercepts, and vertex.

Graph of non linear function, origin O. Horizontal axis, price, dollars, from 0 to 9 by 1’s. Vertical axis, revenue, thousands of dollars, 0 to 1400 by 200’s. Line starts at 0 comma 0, increases until 4 comma 1200 then decreases until 8 comma 0. Passes through 2 comma 900 and 6 comma 900.

In this section, students begin by exploring the effect on their graphs of the different parameters in factored and standard forms of quadratic functions. They recognize the importance of the vertex of a parabola and are introduced to the vertex form of a quadratic expression. Then they use the vertex form to examine how modifying a function can lead to translating the graph in the coordinate plane.

Several optional lessons are available for students to explore deeper or to practice what they have learned. Students look at the effect of the quadratic coefficient and linear term of a quadratic function in standard form, but an optional lesson is available for students curious about the linear coefficient. There is also an optional practice lesson available for revisiting quadratic functions in context and another about using the parameters in a function to determine whether the graph will open upward or downward.

Three parabolas in x y plane. X axis negative 8 to 8, by 2’s. Y axis negative 20 to 30, by 10s. First parabola labeled y equals x squared opens upward with vertex at the origin. Second parabola labeled y equals x squared plus 12 opens upward with a vertex at 0 comma negative 12. Third parabola labeled y equals open parenthesis, x plus 3, end parenthesis, squared opens upward with a vertex at negative 3 comma 0.

In this section, students formally recognize that there are multiple useful forms for quadratic expressions and functions. They examine standard and factored forms and use the distributive property to recognize the same relationship in both forms. At first, they use an area model, familiar from earlier grades, and similar diagrams to do the distribution, but are encouraged later to work without the diagram. 

Then they connect the structure of different forms of the equations to features of the graph, including the intercepts and vertex.