Mathematical Reflection 2

1) You can use a rectangle to model the distributive property. The rectangle shown at the bottom of the post illustrates the expression (x+3)(x+4). When simplifying this expression using the distributive property, you would distribute the x and the 3 separately, so it would look like this: x(x+4)+3(x+4). You would solve this by adding up all the products of your distribution. The model shows all the separate products as parts of a whole. You distribute the 3 and x to the x and the 4, which gives you four separate values, which are represented by the four parts of the rectangle. 2)a. To put an expression in factored form into expanded form, you could distribute. You would take one quantity and distribute the value(s) in it separately to the other quantity. For (x+4)(x+2), you would distribute like this: x(x+2)+4(x+2). b. It is generally easier to work with quadratic expressions when they are in factored form, but that means that there will be times when you have to change from expanded to factored. Expanded form usually looks like this: x²+5x+6. There is sometimes more than one way to write the expression in factored form. First, you have to split up the coefficient of x, and the two values have to multiply to the constant in the expression. 3) A function's equation is quadratic if the highest power of the variable is 2 in standard form, and in factored form the equation must have exactly two linear factors to be quadratic. 4) A quadratic function's graph is always a parabola. In factored form, the opposites of the two constants are the x-intercepts, and the line of symmetry is the value between them. The difference between the constants will tell you if it is opening up or down on the coordinate plane.

Math Reflection p. 39

1. If you have a square, the area is x^2. If you add to the length and width (ex. 3 and 5), you get a new rectangle and therefore a new equation for the area. (x+3)(x+5). The Distributive Property is when you distribute an equation in factored form, (x+3)(x+5) to get a simplified expression in expanded form, x^2+8x+15. The rectangle that has a length of x+3 and a width of x+5 has those two equations: (x+3)(x+5) and x^2+8x+15. You can find the area by using either of these two equations.

2. a) If a quadratic expression is in factored form, you can change it to expanded form by distributing.

factored form: (x+3)(x+5)

You have to multiply the first x by everything in the second set of parenthesis. So you have x*x or x^2 and x*5 or 5x. Now you have to multiply the number in the first set of parenthesis (in this case 3) by everything in the second set of parenthesis. So you have 3*x or 3x and 3*5 or 15. Now your equation in expanded form is x^2+5x+3x+15 or x^2+8x+15.

2. b) If a quadratic expression is in expanded form, you can change it to factored form by splitting each term into smaller terms that will then be able to be distributed back into the same equation in expanded form.

expanded form: x^2+8x+15

The x^2 is easily split into x*x so you know that the factored equation will be (x+?)(x+?). The 8x can be split into 1 and 7, 2 and 6, 3 and 5, or 4 and 4. To know which one it is, you have to figure out the numbers that multiply to 15. 15 can be split into 1 and 15 or 3 and 5. To determine which pair of numbers is correct you have to figure out which one is in both sets. 3 and 5 are in both sets so you know that they will fit into the equation. Therefore, you know what numbers will fit the ?'s. (x+3)(x+5). That is the factored form of the equation x^2+8x+15.

3. You can recognize a quadratic function from its equation by noticing if there is an exponent and if the exponent is a number or a variable. If the exponent is a number then it is a quadratic equation, and if it is a variable then it isn't quadratic. If it is an equation that you can distribute you will have to do that to figure out if there is an exponent and if it is a number or variable.

4. The shape of a quadratic function is always a parabola. Whether it opens up or down, that depends on the equation. A quadratic function usually has two x-intercepts and one y-intercept. To find the x-intercepts, the equation has to be in factored form. You take the opposite of the numbers that are added to x. For example, the equation (x+3)(x+5) would have two x-intercepts, -3 and -5. To find the y-intercept, the equation has to be in expanded form. It is the number added without being multiplied by x. For example, the equation x^2+8x+15 has a y-intercept of 15. Quadratic functions also have a line of symmetry and a maximum or minimum point depending on whether the parabola opens up or down. The equation for the line of symmetry can be found by averaging the two x-intercepts. The x-intercepts of the equation (x+3)(x+5) are -3 and -5 so -3-5=-8/2=-4. Therefore, the line of symmetry is x=-4. To know if the graph has a maximum or minimum point you have to figure out if it opens up or down. The equation x^2+8x+15 has a graph of a parabola opening up. That means it has a minimum point. To find it you can plug the equation for the line of symmetry into the equation to find the y-value of the minimum point. The x-value of the minimum point is the line of symmetry (x=-4 in this case). The line of symmetry is x=-4 so you plug in -4 for the x-values in the equation, x^2+8x+15. -4^2+8(-4)+15=16-32+15=-16+15=-1. Therefore, the minimum point is (-4,-1).

MATH REFLECTION 2

1. The area of a rectangle demonstrates distributive property be multiplying the base by the height. Suppose you have a rectangle with a hieght of x and a base of x + 7. You gat the equation x(x+7). In order to solve this equation you need to distribute and multiply x by x and x by 7. After solving you get x squared + 7x.

2a. When turning quadratic equations from factored to expanded you can use distributive property. The equation (x+3)(x+5) is in factored form. Now to turn it into expanded you will distribute the (x+3) to (x+5). First multiply x(x+5) and then add 3(x+5). You will get x squared +8x +15.

b. When changing expanded form of quadratic equations to the factored form you can use+distributive property. The normal format for an expanded equation is x squared + yx + z. The two numbers that add up to y must also multiply to equal z. Now knowing this you can change x squared + 5x +4. You must find the two numbers that add up to 5 (y) and that also multiply out to equal 4 (z). Once you have found these you take the two numbers and write the equation again but instead of writing y you write ax + bx (a and b are the two number that add up to y and multiply out to z). x squared + x + 4x + 4. You will then distribute one of the x's in x squared to (x+1) and the 4 in 4x and distribute it to (x+1).

3. You can tell whether and equation is quadratic by looking to see if there are any variables that are squared or if there are any of the same variables being multiplied together.

4. Quadratic graphs are parabolas which are lines that look like arches or U's. There will most likely be two x intercepts if the line crosses the x axis. The x intercepts are given to you in the equation, especially if the equation is in factored form. If your equation is y=(x=5)(x-3) your x intercepts are 5,0 and -3,0. The y intercepts is the number that is directly between the two x intercepts. So in this situation it is 0, 1.

Nic S., Joe P., Mike K., and Harry K.

Math Reflection pg 39

1) The area of a rectangle can illustrate distributive property. In expanded form, the equation of a rectangle is simply the added areas of the smaller rectangles that make up the main rectangle. For example, one of the rectangles we examined had an equation A=x^2+10x+25. The same rectangle had a factored equation that looked like this: A=(x+5)(x+5). These equations show that the same rectangle had both A=(x+5)(x+5) and A=x^2+10x+25 for equations. These are equal because the equation A=(x+5)(x+5) distributed is A=x^2+10x+25. This shows what distribution is, simplifying one equation by distributing terms.

2)a.) When an equation is in factored form you can easily find out it's expanded form. You simply distribute. For example: x(x+5)=x^2+5x because by distributing x you can find x*x=x^2 and x*5=5x.

2)b.) You can find an equations factored form by looking at it's expanded form. What you have to do is find a way to split the equation into two mini-equations in parentheses. to do this you split the "X's" into two parts. When split, the X's need to add up to their original value, and multiply to the y-intercept or the number being added to the X's. here's an example:
x^2+13x+12 this equation need to be split into factored form
the 13 x needs to be split
If you split the 13x into 12x and 1x, you have a sum of 13x and when you multiply 12 and 1 you get 12
So you do x(x+1)+12(x+1) or (x+12)(x+1)
that is equal to the original equation x^2+13x+12.

3) You can recognize a quadratic function by its equation because it has a degree no larger than a two.
4) You can tell the y-intercept, xintercept, line of symmetry and minimum or maximum point by looking at a quadratic function's equations. You can find the y-intercept when the equation is in expanded form. It is the number being added or subtracted from the equation. In the equation y=x^2-25 , the -25 is the y-intercept. In the equation y+2x^2+43, 43 is the y-intercept. You can find the x-intercept when the quadratic expression is in factored form. It is the number you add to the summed parentheses to make it equal 0. Often there are two x-intercepts because the parabola of a quadratic function often intercepts the x-axis twice. Here's an examples of finding x-intercepts.
y=(x+5) (x-4) The x-ints are -5 and 4 because you add 5 to -5 to get 0 and 4 to -4 to get 0. It could also be seen as the opposite of whatever number is being added/subtracted from the x.
To find the line of symmetry you average the two x-intercepts. For example, above the x-ints are -5 and 4. -5+4= -1. -1÷2= -1/2. The line of symmetry is -1/2. Then to find the minimum point you plug -1/2 as x into the equation because the line of symmetry's equation is x=-1/2. So to find the minimum point you do y=(-1/2+5) (-1/2-4)=-20.25. -20.25 is the minimum point.

by Kate :-)

Math Reflection

1.a. Graphs and tables of quadratic functions both escalate to a maximum and then decrease symmetrically. The greatest area of a rectangle with a fixed perimeter occurs when the length and width are the same, which appears on the graph as the quadratic function's maximum. A quadratic functions graph also has two x-intercepts.
1.b. In a quadratic function, both the tables and graphs escalate to a maximum and then decrease symmetrically. When x is the length of a rectangle and y is the rectangle's area, and the rectangle has a fixed perimeter, the maximum is the rectangle when the rectangle's length and width are the same. This is an example of a quadratic function.
2. One way to find the maximum area for rectangles with a fixed perimeter is to find the highest point on the parabola representing the quadratic function on a graph. You can also find the maximum area with a table. The greatest area is when the length and width are the same.
3. Tables, graphs, and equations of quadratic functions do not increase or decrease at a steady rate like those for linear functions, nor do they curve exactly like exponential functions. Graphs of quadratic functions always show a parabola, which is unique to quadratic functions.

by Noah S., Peter S., and Ryan F.

Math Refection

1a) The characteristics of graphs of quadratic functions I have observed, so far, have been parabolas. Of tables of quadratic functions, the area goes up until it gets to where the length and width are the same, and then they start to go back down.
1b) Patterns in a graph of a quadratic function appear in the table of values for the function by going up until they get to where the length and width are the same and then go back down to zero.

2) Two ways to find the maximum area for rectangles with a fixed perimeter are to look at the middle area in a table, and to look at the middle of the parabola graph.

3) Tables, graphs, and equations for quadratic functions are very different from those for linear and exponential functions. Tables are different because they don't go up by a specific number or multiplied by a specific number each time. Graphs are because they are not staright lines and they go up and down, not just up. Equations are different because they are not y=mx+b form or y=m to the x+n.

Math Reflection 1 Pg. 18

1a. The graphs I have observed have all been parabolas or upside down U's. Also the graph is symmetrical. The highest point on the graph is always the square. The tables show the shapes facing a different direction. The are on the table increase to a certain point then decrease at the same pattern.

1b. The tables display the pattern on the graph by using coordinate points that go together in table form.

2. One way to find the maximum area for a rectangle with a fixed perimeter is to square the number on the x axis below the highest point on the graph. Another way is to find the largest length on the table and square that.

3. The rate is unsteady in quadratics. The graph is a parabola instead of a line. Also it goes up and down in the same equation.


By, Joe P., Nick S., Harry K. and Mike K.