Quadratic Relations

By: Sonali Guleria

Introduction

The name Quadratic comes from the word quad which means squared. This is because in order for a graph or equation to be a quadratic it needs to be squared e.g.) x². The standard form for a quadratic equation is ax² + b + c=0. Values of A, B and C are given and the values of x are unknown. The graph of a quadratic relation is called a parabola, some of the important features that a parabola has are axis of symmetry, 2 zero's, a vertex, y-intercept and a optimal value.
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Table of Values & First difference to Analyze Quadratics.

How to figure out if the reation is linear or non-linear.

When you look at a table of values, there will be two values given one for independent variable and for dependent viable. The independent variable can be controlled before the trail begins and the dependent variable is measured during the trail, affected by change in the independent variable. You can tell if the relation is linear or non-linear because the points in a non-linear relationship will not lie along a line but form a graph that is curved. If a relation is non-linear or quadratic, the first difference is not constant but the second difference is.

Example: Linear relation

Table of values shows you the relationship between time, in one month intervals, and the height of a plant. Is the relation linear or non-linear?




As you can see from this table of values that, the relation is linear because the first difference is constant.

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Non-linear relation (quadratic)

Use finite differences to determine whether each relation is linear or quadratic?



The relation is quadratic since, the first difference isn't constant but the second difference is.

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Investigating vertex form

Vertex form is converting quadratic form y= ax2+bx+c to a vertex form y=A(x-h)2+k. The "a" affects the vertical starch, the "h" affects the horizontal translation and the "k" represents the vertical translation if a value is negative there is a reflection.

Graphing vertex form

The equation y= a(x-h)2+k tells you your "h" is your x, your "k" is your y and your "a" decides how much your "a" value changes the step pattern when it multiplies with the y-axis point. E.g.) for the equation y= -3(x-3)2-5 you -3 your x and -5 is your y and -3 is the value that changes the step pattern when multiplying with the y-axis. When graphing your x and y your x changes signs, if in the bracket your sign us negative it changes to positive and if in the bracket your sign is positive then it changes to negative. So the final points would be (3, -5) now you would just graph these point ... The final graph is shown in the picture below.
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Finding equation and analyzing vertex form.

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To find the vertex equation for the graph above you need to find the vertex points x and y, which are (5,-3). Then find what number is the step pattern multiplying by with the y-axis, which in this graph is 4. Now you take the values and put them in the vertex equation y= a(x-h)2+k. First you put in the value for "a" which is 4 then for x which is 5 but when you put it into the vertex equation remember the sign changes either from positive to negative or negative to positive but in this case it's positive to negative. Then you put in the value for y which is -3. So the equation would be y= 4(x-5)2-3.

Topic #2

Multiplying Binomials
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Common factors

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Factoring by grouping

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Factoring trinomials with leading 1

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Factoring trinomials - special cases

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Topic #3

Completing the Square
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Solving from Vertex form

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Quadratic formula

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