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 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.

In this section, students examine important parts of a parabolic graph, including the vertex, focus, and directrix. First, students are given quadratic functions in the form and recognize that the vertex of the graph is at the point . Then students use the connection between the vertex and the function in vertex form to write possible quadratic functions from a given graph.

Then students use the idea that points on a parabola are equidistant from a point called the “focus” and a line called the “directrix” to write a function for the parabola. Using the Pythagorean Theorem to find distances on the coordinate plane, students recognize that parabolas must be described by quadratic functions.

An optional lesson is included in this section to give students additional practice adjusting the parameters of a quadratic function in vertex form to see how the adjustments affect the graph.

Focus and directrix on coordinate plane, no grid. X axis from negative 3 to 7. Y axis from negative 3 to 3. Parabola opens upward with vertex at 2 comma negative 1. Points plotted at negative 2 comma 1, 0 comma negative 0 point 5, 2 comma negative 1, 4 comma negative 0 point 5, and 6 comma 1. Focus plotted and labeled at 2 comma 1. Dotted lines drawn from each point to focus. Directrix, horizontal line, y equals negative 3. Vertical lines drawn from each point to directrix.

In this section, students analyze how the different parameters in factored and standard forms of quadratic functions affect their graphs. Students recognize the connection between factored form and the -intercepts of a graph. Students use technology to find the consequences of adjusting the quadratic coefficient and the constant term of an equation in standard form.

Several optional lessons are available for students to go deeper or to practice what they have learned. An optional lesson is available for students curious about adjusting the linear coefficient. There is also an optional practice lesson available for revisiting quadratic functions in context.

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.