Quadratic Functions

This unit builds on Unit 1’s understanding of functions. In order to graph complex equations, a close examination of the library of functions is essential. Visually, the symmetry of a graph, whether with respect to a point or line, can be determined. An equation of a relation can also be algebraically tested for symmetry with respect to an axis or the origin. Using this background information on symmetry, it can be understood whether a function is even or odd. The previously mentioned information lays the foundation for an analysis of the movement of graphs. Graphs can be translated vertically and horizontally as well as reflected and stretched or shrunk. The coefficient of a quadratic function as well as the values of h and k affect the graph of the function, and this information can be summarized in vertex form, y=a(x-h)^2+k. It is also possible to establish if the graph has a maximum or minimum value. By completing the square, vertex form changes to standard form, y = ax^2+bx+c. Comparisons can be made between vertex form and standard form including methods for finding the vertex, axis of symmetry, and solutions/x-intercepts/zeros/roots. The methods for finding solutions to a quadratic equation include factoring, using the quadratic formula, or using rules of exponents and radicals, the focus on Unit 3. It is possible to find solutions to any quadratic equation, regardless of the coefficient and value of the discriminant. In future mathematics and physics courses, applications of quadratic equations will be encountered.

CA Standards Addressed

8.0 Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.

9.0 Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) 2+ c.

Solving Quadratic Equations

Here are some problem written on index cards with answers on the back. You can print it out and use it in stations, give-one-take-one, etc.(right click to save)

 

>Graphing Quadratic Functions

It is not so easy graphing a quadratic function in Standard Form because you have to do some calculations first. So what we want to use is the vertex form. We can get from standard form to vertex form by completing the square. You can always do this for the same reason that you can always solve a quadratic equation using the quadratic formula. From the Vertex Form, you can find the axis of symmetry, vertex, and max/min by just looking at it. No need to do any calculations. There is also the Factored Form that gives you x-intercepts, but that is available only if the function is factorable.

  • Standard Form
  • Vertex Form
  • Factor Form

>Quadratic Equation Word Problems

There are 3 types of quadratic equation word problems.

  1. First is Projectile Motion word problems. They can be written using gravity with height(h) and time(t), path of the projectile motion in height(h) and distance (x), or just a falling object.
  2. Second is Revenue word problems. Revenue can be written as Revenue=Price X Quantity. When foiled, it will give you a quadratic equation. Profit word problems just includes subtracting a constant, which is costs. Profit = Revenue - Cost.
  3. Third is Area problems. Area problems can just involve area or can include a border.

Download handout for Quadratic Equation Word Problems involving Projectile Motion and Revenue.

Worksheets

Download Homework Problems