Final Questions

Solved!

  1. Graph 6x – 2y = 8

    1. Solution

      1. One way is to rewrite the equation into slope-intercept form:

        -6x -6x


        slope-intercept form

      2. Now you know the slope is 3 or and since slope is
        AND you know that the         y-intercept is at (0,-4) (b=-4).


      3. Graph the y-intercept and use the slope to graph the line (up 3 to the right 1 OR down 3 to the left 1).

  2. Graph f(x) =

    1. First realize that f(x) means the function of x is , which simply means that y=. Note: a=1, b=4, & c=3

    2. Now, the axis of symmetry is the invisible line that cuts the parabola in half. The equation of the line for the axis of symmetry is:
      so


      Here is a sketch:DrawObject











    3. Next, solve for the roots of the function:

    4. A graphical organizer way is to use the “Magic X”:

    5. With r1 and r2, make the factors of the quadratic. (x + 3) (x+1).

    6. Because of the zero multiplicative rule (anything times 0 is 0), then the roots (x-intercepts) are when x+3=0 and x+1 = 0

    7. So, the roots are when x = -3 and x=-1 or (-3,0) and (-1,0). Plot those points.

    8. Next, since you know the line of symmetry, the vertex has to be on this line. Therefore, it's x coordinate has to be or x = -2
      Found in step (b) above.

    9. So far we know that the vertex point is at (-2, y). We need the y part, so substitute the x or -2 into the quadratic equation and solve for y:
      substitute x=-2,
      so y=4 – 8 + 3 = -1. Therefore, the vertex point is at (-2, -1). Plot the vertex.

    10. With the roots and the vertex, you can make a sketch of the parabola. If you need further points, substitute x values that look like they would fit the graph and solve for y.

  3. What is the equation of the line that passes through the points (1,3) and (-3,-5)?

    1. Well the slope-intercept form is a great form of the equation. This form is: slope-intercept form: y = mx + b. Where, m is the slope and the b is the y-intercept (where the the line crosses the y-axis).

    2. Next, find the slope of the line that goes through these points. There are a number of ways to do this, however, a way to “think” this out would be:

      1. Roughly plot the points (in your head or a sketch).
        DrawObject



      2. You can see that the slope is either positive (up to the right) or negative (down to the right). In this case it looks positive.

      3. Look at the points' coordinates:



      4. Since slope is rise over run, what is the difference in y values? Well, the difference between 3 and -5 is 8. What is the difference in the x values? Well, it looks like between 1 and -3 is 4.

      5. So now to finalize the slope and it is positive because we knew it was positive slope from 2 (above).

    3. Now, rewrite the equation to y = mx + b (slope-intercept form): y=2x + b.

    4. Last is to find the y-intercept. Just substitute the (x,y) coordinates for one of the points. (1, 3) looks easiest. So: 3 = 2(1) + b. Then,
      3 = 2 + b, 3-2 = b, b= 1.

    5. Voila! The equation of the line is y = 2x + 1.

  4. What are the roots of the quadratic ?

    1. First write down the constants from the equation of the form:
      . In this case: a = 1, b = 2, c = -8 (careful of those negatives!).

    2. Next find the factors of the quadratic equation. A good graphical organizer for finding factors is the use of the “Magic X”.DrawObject



    3. With r1 and r2, make the factors of the quadratic. (x + 4) (x-2).

    4. Because of the zero multiplicative rule (anything times 0 is 0), then the roots (x-intercepts) are when x+4=0 and x-2 = 0.

    5. So, the roots are when x = -4 and x = 2 or (-4, 0) and (2, 0).

  5. Solve

    1. Just like a regular equation, subtract 7 to both sides to “get rid of” the 7 on the left side of the inequality. You will be left with:

    2. Next, to “get rid of” the -2 on the left, we will divide both sides by -2. Note that whenever you divide or multiply by a negative number you must reverse the inequality sign (because, 2 < 3 but -2 > -3). You would be left with:
      your final answer!

  6. Simplify

    1. Use the distributive property to multiply the 5x with the first expression within the parentheses. You will end up with:

    2. Now, be careful! That minus sign between the expression is a hidden -1. I like to make it into an addition problem:

    3. Then, distribute the -1 in the second expression to get:

    4. Combine like terms, which are the expressions with similar exponents on the similar variables. So, add the -10x2 and the -7x2 to get -17x2 and so on. You will be left with:
      your final answer!

  7. Simplify

    1. There is a couple of ways to do this, but first do the old school “flip and multiply” with the fractions. So you would end up with:

    2. Now, we would like you to factor the numerator expression on the first fraction, and “take out” a 3 from the denominator of the first fraction. You would end up with:

    3. Now, cancel all similar expressions in the numerators and the denominators. You would end up with:
      1 ! (not the factorial)

  8. Simplify

    1. Note that a negative exponent means that the variable (or number) with a negative exponent are in the denominator of a fraction. For example, Another example is, Also, be careful of the following, Those are tricky.

      1. A student's trick is to remember that any negative exponent will be in the opposite side (numerator and denomintor) and be made positive. See above for confirmation of this trick.

    2. Also, multiplying the same variable with different exponents will be adding exponents, since,

    3. Taking a variable with an exponent to an additional exponent will be multiplying the exponents,

    4. So, using the above notes, simplify the expression and cancel similar variables as needed.
      Now cancel the z's and combine the y's, so that your answer is

  9. (Please note that this is a different question and reflects a more correct question that will be found on the Final for TEAM ALPHA)
    Arneson wanted to rent a bicycle at the beach . It costs $25 for a rental fee and $8 for each hour of use. If he paid $61 for his bicycle rental, how many hours did he rent the bicycle?

    1. Write an equation that you could use to solve this problem.
      Givens: (1) cost of rental fee (one time fee) = $25, (2) Cost per hour = $8, (3) total paid at the end of day = $61
      Unknowns: (1) How many hours did he rent it? I choose the variable for this unknown to be h.
      E      quations: 8h + 25 = 61 ($8 per hour plus the rental fee of $25 is $61)
      Substitute: Already done that!

    2. Solve the problem

      8h + 25 = 61

      - 25 -25

      8h = 36

      8h = 36
      8 8

      h = 4

      So, he rented the bicycle for 4 hours!













  10. Altogether 292 tickets were sold for a school play. An adult ticket costs $3 and a student ticket costs $1. The ticket sales totaled $470. How many student tickets were sold?

    1. Write a system of equations that you could use to solve this problem.
      Givens: (1) Total tickets sold = 292, (2) Adult ticket price = $3, (3) student ticket price = $1, (4) total sales in dollars = $470
      Unknowns: Number of adult tickets: I choose a variable of a, and the number of student tickets: I choose a variable of s.
      E      quations:
      the number of adult and student tickets total 292.
      the number of adult tickets times the cost per ticket ($3) plus the number of student tickets times the cost per student ticket totals $470.
      Substitute: Already done that!

    2. Solve the problem
      Elimination or linear combination method:
      I see that the       s variables cancel if I subtract the equations straight down. This will leave me with a in the equation:

      a + s = 292
      -(3a + s = 470)
      -2a +0 = -178

      -2a = -178
      -2 -2

      a = 89










      There are 89 adult tickets! Now, substitute this number of adult tickets into the first equation (which is much easier) and solve for s:




      a + s = 292 , Now

      89 + s = 292

      -89 -89

      s = 203








      So, there are 203 student tickets! Final answer is there are 203 student tickets (that is what the question is only asking for!).




      Now for the substitution method:
      Use the easier first equation and solve for a:

      a + s = 292
      -s -s

      a = 292 - s









      Now, substitute this new equation into the second equation and solve for s:


      a = 292 -s

      3a + s = 470, So,

      3 (292-s) + s = 470

      distribute

      876 – 3s +s = 470

      add like terms

      876 – 2s = 470

      -876 -876

      -2s = -406
      -2 -2

      s = 203





















      Once again, the final answer is the number of student tickets is 203!

      WHEW!