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Instructions on how to find the vertex of a parabola, graph a quadratic function, use the discriminant to determine the number of x-intercepts, and solve problems involving maximum or minimum values. It also covers parabolas with horizontal axes and completing the square. Written by Cindy Alder.
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Written by: Cindy Alder
Quadratic Equations
๏ We can tell a lot about a quadratic equation when it is in the form:
๏Vertex: ๏Axis of Symmetry: ๏Direction of opening: UP if DOWN if ๏Wide/Narrow: WIDE if NARROW if ๏Vertical Shift: ๏Horizontal Shift:
axis of symmetry x = h
Vertex ( h, k )
Complete the Square when ๐ = ๐
๐ ๐ = ๐๐^ โ ๐๐ + ๐
Find the Vertex of the Graph of
๐ ๐ = ๐๐^ + ๐๐ + ๐๐
Complete the Square when ๐ โ ๐
Find the Vertex of the Graph of
๐ ๐ = ๐๐๐^ โ ๐๐๐ + ๐๐
The Vertex Formula
๐ ๐ = ๐๐๐^ + ๐๐ + ๐
Vertex Formula
โ๐ ๐๐
โ๐ ๐๐ ๐ = ๐^ ๐๐๐ ๐๐๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐ ๐
Find the vertex using the vertex formula.
๏ ๐ ๐ = ๐๐^ + ๐๐ + ๐๐
Graphing a Quadratic Function
๏ STEPS ๏ Does the graph open up or down? ๏ Find the vertex. ๏ Vertex formula ๏ Complete the square ๏ Find any intercepts. ๏ y- intercept: set ๐ฅ = 0 ๏ x- intercept: set ๐ฆ = 0 ๏ Complete the graph. ๏ Plot points found. ๏ Plot additional points as needed, using symmetry.
_A minimum of two points on either side of the vertex is needed for an accurate graph._*
Graph the Quadratic Equation:
๐ ๐ = ๐๐^ โ ๐๐ + ๐
๏จ Step 1 - Does the graph open up or down?
+ +
๏จ Step 4 - Complete the graph. ๏ Plot points found. ๏ Vertex: 3, โ ๏ y- intercept: 0, ๏ x- intercepts: 5,0 ๐๐๐ (1,0)
๏ Plot additional points as needed, using symmetry. ๏ Using symmetry the graph will also pass through the point 6,.
Graph the Quadratic Equation (continued)
๐ ๐ = ๐๐^ โ ๐๐ + ๐
Graph the Function ๐ ๐ = ๐๐^ โ ๐ โ ๐
Using the Discriminant
x -intercepts of the graph of each quadratic function.
Parabola with Horizontal Axis
๐ = ๐๐๐^ + ๐๐ + ๐ or ๐ = ๐ ๐ โ ๐ ๐^ + ๐
is a parabola with vertex (โ, ๐) and the horizontal line ๐ฆ = ๐ as axis of symmetry. The graph opens to the right if ๐ > 0 and to the left if ๐ < 0.