Tag Archives: quadratics

Week 13 blog post

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On this week, we had a team work activity, we graph and solve absolute value functions and equations at the whiteboard. This activity was very fun. We discussed with our team and we helped each other. It was good at teamwork. Also we learned Graphing Reciprocals of Linear functions and Graphing Reciprocals of Quadratic functions. In these parts, the equation is looked like y= \frac{1}{x} Its parents function is y= x. The reciprocal function’s form is to reflect the parents function. For example, y= x+5 Its reciprocal  function is y= \frac{1}{x+5}. In reciprocal functions, there are some important things, the graph of y= \frac{1}{x-1} has no x-intercept because latex \frac{1}{x-1}$ is undefined when x=1. That is, x=1 is a non-permissible value. The line x=1 is a Vertical Asymptote; that is, a vertical line that the graph approaches but never reaches. This asymptote passes through the x-intercept of y=f(x). In a word, Horizontal asymptote is y=0, Vertical asymptote is x-intercept. When we are graphing the reciprocal function, we have to know invariant points. Invariant points is the points when the y is 1 and -1. In linear reciprocal functions, there is one vertical asymptote. But in quadratic functions, there are maximum of two vertical asymptote. When we graphing Reciprocal functions, first we have to graph parents function and to find the points when y= 1 and -1, and finally to find vertical, horizontal asymptote. In linear reciprocal functions, the invariant points is maximum of two but in quadratic reciprocal functions , the invariant points is maximum of four.

 

Graphing Linear Reciprocal function and Quadratic function.

Week 11 blog post

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On this week, we learned how to graph Linear inequality. First, we have to check y-intercept and slope. and next, we have to check inequality sign. If any equation has equal sign, we have to draw a solid line, however the equation doesn’t have equal sign, we have to draw broken line. And if we graph the linear inequality, the graph of the equation divides the coordinate plane into two region, and to put the test number into the equation. and if the equation include the part of to include the test number, we have to paint that part.

Ex) Determine whether each point is a solution of the given inequality

4x-3y ≥ 6     A(4,3)       4(4)-3(3) ≥ 6 ->   16-9 ≥ 6   ->   16 ≥ 15    This point is solution.

7y+2x < -10   B(-8,1)     7(1)+2(-8) < -10 ->  7-16 < -10  ->  -9 < -10   This point is not solution

-y+5x ≤ 17     C(2,-7)        -(-7)+5(2) ≤ 17 ->  7+10 ≤ 17 -> 17 ≤ 17  This point is solution

-3x+8y+5 > 0 D(-5,-3)    -3(-5)+8(-3)+5 > 0 ->   20-24 >0 ->  -4 > 0 This point is not solution

Also on this week, we learned how to graph Quadratic inequality. First, we have to find the coordinates of vertex, slope(a), x-intercepts, y-intercept,and direction of opening. And check the inequality sign of the quadratic equation. If the equation has equal sign, we have to draw a solid curve, if the equation doesn’t have equal sign, we have to draw a broken curve. And next, to put the test number into the equation.(Test points can be used to determine which region satisfies the inequality.) And also we have to paint the part of to include the test number.

Ex) To graph Quadratic inequality

 

Week 9 blog post

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On this week , we finished lesson 4.5 -4.7 and we reviewed midterm.

On these lessons, we learned how to change the equation to completing the square from y= ax^2+ bx +c, a > 0, we learned to find x-intercept by to factor. And we learned to graph a quadratic function from its equation in general form, also we learned to use a quadratic function to model a situation.

Ex) Write each equation in standard form

y =x^2 + 4x +8                                           y= 2x^2 +16x -10

y= (x^2 +4x +4-4)+8                                y= 2(x^2 +8x+16-16)-10

y=(x+2)^2 -4+8                                         y= 2(x+4)^2-32-10

y= (x+2)^2 +4                                            y= 2(x+4)^2-42

Ex)  Identify the x-intercepts of the graph of each quadratic function.

a) y= (x-12)(x+11)                                          b) y= (x+5)(x-17)

x-intercept : 12, -11                                    x- intercept : -5, 17

c) y=(2x+2)(x-6)                                            d) y= x(x-9)

x-intercept : -1, 6                                            x-intercept : 0,9

Ex) Determine the intercepts, the equation of the axis of symmetry, and the coordinates of the vertex of the graph of each quadratic function.

a) y= $latex 2x^2 +16x+24

y= 2(x^2 +8x+12)

y= $latex 2(x+6)(x+2)

x-intercept : -6, -2

y-intercept : 24

y= 2(x^2 + 8x+16-16)+12

y= $latex 2(x+4)^2-32+12

y= $latex 2(x+4)^2-20

Vertex : (-4, -20)

axis of symmetry : x=-4

Ex) A rectangular area is divided into 2 rectangles with 200 m of fencing used for the perimeter and the divider. What are the dimensions of the total maximum area.

3W+2L=200

2W/3 + L = 100

L = 100 – 2W/3

 

W * L = Maximum

 

W(100-2W/3)=Maximum

100W-2W^2/3 = Maximum

-2/3($latex W^2 – 200/3+40000/36 -40000/36)=Maximum

-2/3(W-200/6)^2+ 80000/108= Maximum

-2/3(W-200/6)^2+20000/27= Maximum

W = 100/3m

L = 50m

 

 

 

 

 

Week 8 blog post

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On this week, we learned new chapter. The new chapter is Analyzing Quadratic Functions. In this chapter, we learned how to draw the graph. The basic equation is y= a(x-p)^2 +q. In this equation, a means slope and if a is positive, it opens up and if it is negative, it opens down. Also when the graph is opens up, it has minimum vertex, and when the graph is opens down, it has maximum vertex. Also if a is 1, it always congruent with y= x^2. The next thing, p means horizontal translation, q means vertical translation. And x means x-intercept, y means y-intercept. And also if  absolute value a is bigger than zero, it is stretch and absolute value a is less than zero, it is compression. If we know these things, we can know how to be transformed at y= x^2.

Ex) The graph of y= x^2 is translated as described below. Without graphing, write the equation of the graph in its new position.

a) a translation of 16 units up and 13 units right                                b) a translation of 33 units down and 29 units left.

y= (x-13)^2 + 16                                                                  y= (x+29)^2 – 33

Also we have to know about the axis of symmetry. The axis of symmetry is a line through a shape so that each side is a mirror image. When the shape is folded in half along the axis of symmetry, then the two halves match up.

Ex) To find the axis of symmetry

a) y= (x-19)^2                   b) y= (x+7)^2                   c) y= x^2 +9

x=19                                                            x=-7                                                    x=0

EX) To find vertex

a) y= 5(x+7)^2 -13                    b) y= 0.164(x-26)^2+3

(-7,-13)                                                             (26,3)