Math 2

​Math 2

By Joanna Li

Translating Words to Math

Translating Words to Math

Example 1: If 8 less than twice a number equals 4 plus the product of 6 and 3, what is the number?

Multiple Choice

The product of a number and 2.4 is 0.48. What's the mathematical translation to this problem?

• 0.48x=2.4

• 2.4x=0.48

• 2.4(0.48)=x

Multiple Choice

The difference of 2 times a number and 3/4 is 1/2 . What's the mathematical translation for this problem?

• $$2x-\frac{3}{4}=\frac{1}{2}$$

• $$\frac{3}{4}-2x=\frac{1}{2}$$

• $$2x=\frac{3}{4}-\frac{1}{2}$$

Multiple Choice

Tommy runs 2m miles per day for p days. He then runs m miles for an additional q days. In terms of m, p, and q, how many miles did Tommy run altogether?

• 2mp+mq

• 3m

• 2m(p+q)

• 2m+mq

Algebraic Operations

• Monomial: Expression with only 1 term

  • i.e., 476, 4z, n

• Coefficients: Numbers in front of the variable in the expression

  • i.e., In the term 4z, 4 is our coefficient

Multiple Choice

What is the coefficient of the monomial $$z$$ ?

• 0

• 1

Operations with Monomials

Multiple Choice

Simplify this expression $$2x^2+x\left(1+2y^3\right)$$

• $$2x^2+x+2xy^3$$

• $$2xy^3+3x$$

• $$2x^4y^3$$

Single Variable Equations

Single Variable Equations

Multiple Choice

What is the value of x if $$\frac{\sqrt[]{x}}{2}=4$$

• 4

• 8

• 16

• 64

Multiple Choice

What's the value of x if $$x^2-9=0$$

• $$\pm3$$

• -3

• 3

• 9

Multiple Choice

What is the value of x if $$\frac{7}{8}x=14$$ ?

• 14

• 16

• 20

• 24

Multiple Choice

If $$6x-2+3x=9-4x$$ , then what is the value of $$26x+2$$ ?

• 4

• 6

• 12

• 24

Linear Equations

Multiple Choice

What is the slope and y-intercept of the equation $$-\frac{2}{5}x=y+3$$

• $$m=-\frac{2}{5},\ b=-3$$

• $$m=-\frac{5}{2},b=-3$$

• $$m=\frac{5}{2},b=3$$

• $$m=\frac{2}{5},b=3$$

Linear Equations

Multiple Choice

What is the slope of a line containing the points $$\left(3,6\right),\left(-2,5\right)$$

• 1/5

• -1/5

• 1/10

• -1/10

Multiple Choice

If the slope of the line containing points A(-6,3) and B(5,w) is 2, what is the value of w?

• -2

• 0

• 25

• 19

Linear Equations

Multiple Choice

Does the following line have a positive, negative, zero, or undefined slope?

• positive

• negative

• zero

• undefined

Linear Equations

Multiple Choice

What is the equation of the following line?

• x=4

• y=4

• y=5

• x=5

Linear Equations

Multiple Choice

What is the equation of the following line?

• x=1

• x=5

• x=3

• y=1

Parallel and Perpendicular Lines

Multiple Choice

What is the slope of a line perpendicular to $$y=-2x$$

• $$\frac{1}{2}$$

• 2

• -2

• $$-\frac{1}{2}$$

Linear Equations

Linear Equations

Multiple Choice

What is the x-intercept of $$2x+2y=8$$ ?

• 0

• 1

• 2

• 4

Multiple Choice

What is the y-intercept of $$y=2x^2+4x+2$$ ?

• 0

• 1

• 2

• 4

Linear Equations

Linear Equations

Linear Equations

Multiple Choice

What is the equation of the line that goes through (1,1) and (0,-1)?

• y=2x-1

• y=-1/2x

• y=-2x-1

• y=2x

Multiple Choice

What's the equation of the line that goes through (1,2) and (1,3)?

• x=1

• y=1

• y=x

Multiple Choice

What's the equation of the line that has a y-intercept of 2 and an x-intercept of 3?

• y=-2/3x+2

• y=-2/3x+3

• y=2/3x+2

• y=2/3x+3

Linear Equations

2. Plug in your slope into the point-slope formula, then plug in your point to solve for the y-int

Multiple Choice

Which of the following equations defines a line in the xy-plane that has a slope of 1/6 and passes through the point (12, -7)?

• $$y=12x-7$$

• $$y=\frac{x}{6}-9$$

• $$y=\frac{x}{6}-7$$

• $$y=-9x+\frac{1}{6}$$

Multiple Choice

If a linear function h satisfies h(-1) = 3 and h(0) = 5, what is the equation of the function?

• $$h\left(x\right)=2x+5$$

• $$h\left(x\right)=3x+5$$

• $$h\left(x\right)=2x+3$$

• $$h\left(x\right)=2x$$

Multiple Choice

Which equation defines linear function g with a slope of −13 that passes through the origin?

• $$g\left(x\right)=-13x$$

• $$g\left(x\right)=13x$$

• $$g\left(x\right)=-\frac{1}{13}x$$

• $$g\left(x\right)=x-13$$

Side Note: Roots, Zeros, and Solutions

Side Note: Roots, Zeros, and Solutions

Introduction to Quadratics

Multiple Choice

What is the x-intercept(s)?

• -1,3

• 1,-3

• 0

Multiple Choice

What is the solution(s) to this parabola?

• 1

• -1

• 0

Prerequisite: FOILing

Multiple Choice

Which of the following is equivalent to $$\left(x+3\right)\left(x-4\right)$$

• $$x^2-x-12$$

• $$x^2+x-12$$

• $$x^2-x+12$$

Multiple Choice

Which of the following is equivalent to $$\left(x+2\right)^2$$

• $$x^2+4x+4$$

• $$x^2-4x-4$$

• $$x^2+2x+4$$

• $$x^2+4x+2$$

Tool #1 — Factoring & Reverse FOIL

Reverse FOIL Example

Multiple Choice

Which of the following equations are equivalent to $$x^2-7x+10$$

• $$\left(x-2\right)\left(x+5\right)$$

• $$\left(x+2\right)\left(x-5\right)$$

• $$\left(x+7\right)\left(x+10\right)$$

• $$\left(x-7\right)\left(x+10\right)$$

Multiple Choice

Which of the following equations are equivalent to $$x^2+2x=8$$

• $$\left(x+4\right)\left(x-2\right)$$

• $$\left(x-4\right)\left(x+2\right)$$

• $$\left(x+2\right)\left(x+2\right)$$

• $$\left(x+4\right)\left(x-4\right)$$

Reverse FOIL Example: Box Method

Multiple Choice

Which of the following equations are equivalent to $$x^2+5x=-6$$

• $$\left(x+2\right)\left(x+3\right)$$

• $$\left(x+2\right)\left(x-3\right)$$

• $$\left(x-2\right)\left(x+3\right)$$

• $$\left(x-2\right)\left(x+3\right)$$

Multiple Choice

Which of the following is equivalent to $$2x^2+7x+3$$

• $$(2x+1)(x+3)$$

• $$(2x-1)(x+3)$$

• $$(2x+1)(x-3)$$

• $$(2x-1)(x-3)$$

Why Factoring Works

Tool #1 — Factoring & Reverse FOIL

Multiple Choice

What are the solutions to $$x^2+6x+9$$

• $$\pm3$$

• $$\pm6$$

• $$\pm9$$

Multiple Choice

What are the solutions to $$9x^2-12x+4$$

• $$\pm\frac{2}{3}$$

• 2/3

• -2/3

Multiple Choice

What are the solutions to $$x^2-16=0$$

• 4

• -4

• $$\pm4$$

Multiple Choice

What are the solutions to $$4x^2+9$$

• $$\pm3$$

• $$\pm\frac{3}{2}$$

• $$\pm\frac{2}{3}$$

Factor before Reverse FOIL

Factor before Reverse FOIL

Factor Before Reverse FOIL

Multiple Choice

What is the solution to $$2x^2-8x-10$$

• 5,-1

• -5,1

• 5,1

• -5,-1

Tool #2 — The Quadratic Formula

Multiple Choice

How many solutions does $$y=x^2+2x+3$$ have?

• 0

• 1

• 2

Multiple Choice

What are the solutions of $$z^2-7=18$$

• $$\pm5$$

• $$\pm6$$

• $$\pm4$$

• $$\sqrt[]{25}$$

Vertex of a Quadratic

Multiple Choice

What is the vertex of the function $$f\left(x\right)=-2x^2+10x-12$$

• (-2,-3)

• (0,-12)

• (2,3)

• (2.5,0.5)

Multiple Choice

$$y=a\left(x+4\right)\left(x-2\right)$$

In the quadratic equation above, a is a non-zero constant. The graph of the equation in the xy-plane is a parabola with vertex (c,d). Which of the following is equal to d?

• -1

• 2

• -a

• -9a

Concavity of a Quadratic

Multiple Choice

Is the following quadratic concave up or down?

• concave up

• concave down

Multiple Choice

Is the following function upward- or downward-facing? $$y=x^2+2x+3$$

• upward facing

• downward facing

Standard, Vertex, Intercept Form

Multiple Choice

What is the vertex of $$y=2\left(x-4\right)^2+3$$

• (4,3)

• (-4,3)

• (4,-3)

• (-4,-3)

Multiple Choice

What is the vertex of $$y=-\left(x-3\right)^2-2$$

• (3,-2)

• (-3,-2)

• (-3,2)

• (3,2)

Multiple Choice

$$y=−x^2+k$$

In the equation above, $$k$$ is a positive constant and the graph of the equation in the xy-plane is a parabola. Which of the following is an equivalent form of the equation?

• $$y=-(x-\sqrt{k})(x+\sqrt{k})$$

• $$y=-(x+k)(x+k)$$

• $$y=(x+k)(x-k)$$

• $$y=-(x+k)(x-k)$$

Multiple Choice

$$y=a(x+6)(x-2)$$

In the quadratic equation above, $$a$$ is a nonzero constant. The graph of the equation in the xy-plane is a parabola with a vertex of (h, k). Which of the following is equivalent to $$k$$ ?

• -16a

• 0

• -4a

• -12a

Multiple Choice

$$y=\left(x-5\right)\left(x+4\right)$$

Which of the following is an equivalent form of the equation above in which the minimum value of y appears as a coefficient or constant?

• $$y=x^2-x-20$$

• $$y=\left(x-0.5\right)^2-20.25$$

• $$y=-\left(5-x\right)\left(x+4\right)$$

• $$y=-\left(x-5\right)^2+4$$

Multiple Choice

$$y=\left(x+5\right)^2-4$$

What is the distance between the solutions to the quadratic above?

• -4

• 0

• 4

• 10

Algebraic Inequalities

Number Line Graphs

• We can depict the solution set of an inequality by graphing on a number line

• An open circle indicates that the value is not a possible solution

• A closed circle indicates that the value is a solution

Multiple Choice

What is the inequality that is graphed

• x>8

• x<8

• $$x\ge8$$

• $$x\le8$$

Solving for Algebraic Inequalities

• Same exact process as solving for algebraic equalities

• Exception: When multiplying/dividing by a negative number, flip the direction of your sign

• Remember that inequalities cover a range of values, not just one

Multiple Choice

If $$3a+4>6a-5$$ , what is the complete range of values of a?

• $$a<1$$

• $$a>.333$$

• $$a<3$$

• $$a>3$$

System of Equations

• Algebraic problem involving multiple equations and variables

  • Intersection point is the solution for the variables

  • Usually focuses on system of linear equations, which can have 0, 1, or infinite solutions

System of Equations

​System of Equations

System of Linear Equations

2. Elimination: Eliminating a variable by combining your 2 equations (or a factor of one or more of your equations)

Multiple Choice

If $$7x+8y=12$$ and $$2x+2y=28$$ , then $$x=?$$

• 1

• 10

• 100

• 1000

Multiple Choice

A coffee shop sells two types of coffee: arabica and robusta. The owner creates a new blend by mixing x pounds of arabica beans with y pounds of robusta beans. Arabica beans cost $8 per pound, and robusta beans cost $6 per pound. The final blend weighs 20 pounds and costs $140. Which system models this?

• x + y = 20, 8x + 6y = 140

• x + y = 140, 6x + 8y = 20

• x + y = 20, 6x + 8y = 140

• x + y = 140, 8x + 6y = 20

Multiple Choice

Gloria purchased notebooks and markers for five classes in her school. The notebooks cost $1.50 each and the markers cost $2.25 each. Her receipt tells her that she purchase 30 total items and that her total cost was $55.50. Which system of equations could be used to figure out how many notebooks, n, and markers, m, she purchased? Assume there was no tax.

• $$m+n=55.5$$

  $$2.25m+1.5n=30$$

• $$m+n=30$$

  $$2.25m+1.5n=55.5$$

• $$5\left(m+n\right)=55.5$$ $$2.25m+1.5n=30$$

• $$5\left(m+n=55.5\right)$$ $$2.25m+1.5n=30$$

Variable Relationships

• Questions where you're asked to solve for one variable in terms of another

• Isolate the variable asked to solve for, then leave everything else on the other side

Multiple Choice

If $$4x+7=xy-6$$ , what is the value of x, in terms of y?

• $$-52-13y$$

• $$4y-16$$

• $$-\frac{13}{4-y}$$

• $$\frac{1}{4-y}$$

Multiple Choice

If $$k+3=7-2y$$ and $$k=8-2p$$ , what does y equal in terms of p?

• $$p-2$$

• $$p+6$$

• $$p+2$$

• $$2p+12$$

Advanced System of Equations

• Sometimes, a system may have one linear and one quadratic

• Envision any nonlinear system of equations graphically to determine the possibilities

  • Graphically, our solution represents all point(s) of intersection

  • Algebraically, the variables that satisfy all equations in your system

• A linear-quadratic system may have 0, 1, or 2 solutions

Multiple Choice

$$y=mx^2+n$$

$$y=4$$

In the system of equations shown above, m and n are constants. For which of the following values of m and n does the system have no real solutions?

• m=-3,

  n=5

• m=-3,

  n=4

• m=5,

  n=4

• m=3,

  n=6

Multiple Choice

$$y=7x−k$$

$$y=3x^2+37x+84$$

The system of equations has exactly one real solution at (x,y). If k is a constant, what is the value of x?

• 9

• 5

• -5

• -9

Multiple Choice

If $$y=x^2+11x+15$$ and $$y=3x-1$$ , then what is the value of x?

• -16

• -4

• 3

• 4

Multiple Choice

$$y=x+4$$

$$z=\frac{1}{2}y^2+3$$

Given the system of equations above, which of the following is equivalent to $$3x+3z$$

• $$y-1$$

• $$y+1$$

• $$y^2+3y-6$$

• $$y^2-3y+18$$

Multiple Choice

$$y=x^2−6x−18$$

$$y=−x+6$$

A solution to the system of equations is (a,b). If a<0, what is the value of a?

• 8

• -3

• -6

• -18

Multiple Choice

$$x^2-2xy+y^2=144$$

$$y=3x$$

In the system of equations above, if (x,y) is a solution, then what is the value of $$y^2$$

• 6

• 18

• 36

• 324

System of Inequalities

• Solution sets involve regions of space

  • Algebraically, all points that satisfy all inequalities in the system

Multiple Choice

A publishing company is printing two types of books: paperbacks and hardcovers. Each paperback takes 3 hours to print and 2 hours to bind. Each hardcover takes 4 hours to print and 5 hours to bind. The printing department has a total of 18 hours, and the binding department has a total of 22 hours available each day.

Let $$x$$ be the number of paperbacks and $$y$$ be the number of hardcovers produced in a day. Which of the following systems of inequalities represents the constraints on production?

• $$3x+4y\le22$$

  $$2x+5y\le18$$

• $$3x+4y\le18$$

  $$2x+5y\le22$$

• $$3x+2y\le22$$

  $$4x+5y\le18$$

• $$3x+2y\le18$$

  $$4x+5y\le22$$

Multiple Choice

$$y=2x^2-3x+4$$

$$y=rx-2$$

If the x-values to the system of equations above are x=1.5 and x=2, what is the value of r?

• 1

• 2

• 3

• 4