Graphing Quadratic Functions from Factored Form

 
Videos and solutions to help Algebra I students learn how to use the factored form of a quadratic equation to construct a rough graph, use the graph of a quadratic equation to construct a quadratic equation in factored form, and relate the solutions of a quadratic equation in one variable to the zeros of the function it defines.

Students understand that the number of zeros in a polynomial function corresponds to the number of linear factors of the related expression and that different functions may have the same zeros but different maxima or minima.

New York State Common Core Math Module 4, Algebra I, Lesson 9

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Lesson 9 Summary

When we have a quadratic function in factored form, we can find its x-intercepts, y-intercept, axis of symmetry, and vertex.

For any quadratic equation, the roots are the solution(s) where y = 0, and these solutions correspond to the points where the graph of the equation crosses the x-axis.

A quadratic equation can be written in the form y = a(x - m)(x - n), where m and n are the roots of the quadratic. Since the x-value of the vertex is the average of the x-values of the two roots, we can substitute that value back into equation to find the -xvalue of the vertex. If we set x = 0, we can find the y-intercept.

In order to construct the graph of a unique quadratic function, at least three distinct points of the function must be known.

Examples:
f(x) = x² + x - 30, f(x) = -(x + 2)(x -5)
Find the x-intercepts, y-intercept, axis of symmetry and vertex or turning point.

Lesson 9 Problem Set Sample Solutions

1. Graph the following on your own graph paper and identify the key features of the graph.

a) f(x) = (x - 2)(x + 7)
b) g(x) = -2(x - 2)(x + 7)
c) h(x) = x² - 16
d) p(x) = x² - 2x + 1
e) q(x) = 4x² - 20x + 24