Given a quadratic equation of the form ax² + by² + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and graph the equation. [In Mathematics III, this standard addresses only circles and parabolas.]
Your teacher will assign one of these equations to your group:
Rewrite your equation in vertex form by completing the square.
Identify the transformations from the equation to your equation.
Before graphing, identify the vertex and -intercept.
Graph your equation.
Student Lesson Summary
When we have an equation for a parabola in vertex form, we can see the transformations from an original function without graphing. Here is an example:
The graph of has been shifted left 6, stretched vertically by a factor of 4, and shifted down 7. This makes sense because the original vertex is at , and the new vertex is at , so it has been shifted left 6 and down 7 as well.
We can also see the transformations from an equation that is not written in vertex form, but we will need to rewrite it first. Take a look at this equation: . Let's rewrite it in vertex form by completing the square:
Now we can see that the vertex is at . Using this equation, we can identify the transformations from : shift left 5, vertical stretch by a factor of , shift down 6.
For any equation of a parabola in vertex form , we can identify the transformations: horizontal translation by , vertical stretch by a factor of , reflection over the -axis if , and vertical translation by .
Glossary
None
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Given a quadratic equation of the form ax² + by² + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and graph the equation. [In Mathematics III, this standard addresses only circles and parabolas.]
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
