Mathematics Concepts Review

This set of flashcards covers various mathematical concepts including linear functions, systems of equations, quadratic functions, polynomials, rational expressions, exponential functions, logarithms, sequences, probability, statistics, and basic trigonometry.

What is the slope of the line given by the equation y = 3x − 7?

The slope is 3.

What is the y-intercept of the line given by the equation y = −2x + 5?

The y-intercept is 5.

Write the equation of a line with slope 4 passing through the point (1, −2).

y = 4x − 6.

If a taxi charges $5 plus $2 per mile, what is the equation describing the cost?

y = 2x + 5.

What is the slope between the points (2, 3) and (6, 11)?

The slope is 2.

When solving the system of equations y = 2x + 1 and y = x + 4, what are the values of x and y?

x = 3, y = 7.

For the system of equations 2x + y = 7 and x − y = 1, what are the solutions for x and y?

x = 8/3, y = 5/3.

What does it mean if two lines never intersect in a system of equations?

It means there is no solution.

If movie tickets cost $10 for adults and $6 for kids, what is another combination that adds up to $38 for 2 adults and 3 kids?

3 adults and 2 kids.

Solve the equations x + y = 10 and x − y = 4.

x = 7, y = 3.

What is the factored form of x² + 5x + 6?

(x + 2)(x + 3).

What are the solutions for the equation x² − 9 = 0?

x = ±3.

Find the vertex of the quadratic function y = x² − 4x + 1.

The vertex is (2, −3).

If a ball is thrown upward described by the equation h = −16t² + 32t, when does it hit the ground?

It hits the ground at t = 2 seconds.

How many real solutions does the equation x² + 1 = 0 have?

It has 0 real solutions.

What is the result of adding (3x² + 2x) + (x² − 5x)?

The result is 4x² − 3x.

What is the product of (x + 2)(x − 3)?

The product is x² − x − 6.

What is the degree of the polynomial 5x³ − 2x + 1?

The degree is 3.

Factor the expression x² − 16.

The factors are (x − 4)(x + 4).

What is the expanded form of the volume of a box given by (x)(x+2)(x−1)?

The expansion gives x³ + x² − 2x.

Simplify the expression (x² − 4)/(x + 2).

The simplification results in x − 2.

What restrictions apply for the function 1/(x − 5)?

x cannot equal 5.

Simplify the expression 2/x + 3/x.

The simplification gives 5/x.

If distance d is given by d/t and d = 120 with t = 3, what is the speed?

The speed is 40.

Solve the equation 1/x = 1/4.

The solution is x = 4.

What is the result of simplifying 2³ × 2■?

The result is 2■ = 128.

What is the solution for the equation 3^x = 9?

The solution is x = 2.

If a population doubles every year, what is the general equation representing this?

The equation is y = a(2^t).

What do you get when you simplify (5²)³?

The simplification is 5■.

What is the solution for the equation 4^x = 1/4?

The solution is x = −1.

Evaluate log■■(100).

The evaluation results in 2.

Rewrite log■(27) = x in exponential form.

The exponential form is 3^x = 27.

If log■(x) = 5, what is the value of x?

x = 32.

What is log■■(1000)?

The result is 3.

Simplify ln(e³).

The simplification gives 3.

What is the next term in the sequence: 2, 4, 6, 8?

The next term is 10.

What is the common difference in the sequence: 5, 9, 13?

The common difference is 4.

Write the formula for an arithmetic sequence where the first term a■=3 and the common difference d=2.

The formula is a■ = 3 + 2(n−1).

What is the slope of the line given by the equation y = 3x - 7?

The slope is 3.

Method: Identify the slope m using the slope-intercept form y = mx + b. Here, m = 3.

What is the y-intercept of the line given by the equation y = -2x + 5?

The y-intercept is 5.

Method: In the equation y = mx + b, the constant b represents the y-intercept. Here, b = 5, so the point is (0, 5).

Write the equation of a line with slope 4 passing through the point (1, -2).

y = 4x - 6.

Method: Use the point-slope formula: (y - y1) = m(x - x1).

1. (y - (-2)) = 4(x - 1)
2. y + 2 = 4x - 4
3. Subtract 2 from both sides: y = 4x - 6.

If a taxi charges $5 plus $2 per mile, what is the equation describing the cost?

y = 2x + 5.

Method: Use the linear formula y = mx + b.

• m (rate) = 2
• b (initial flat fee) = 5.

What is the slope between the points (2, 3) and (6, 11)?

The slope is 2.

Formula: m = \frac{y2 - y1}{x2 - x1}.
Step: m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2.

When solving the system of equations y = 2x + 1 and y = x + 4, what are the values of x and y?

x = 3, y = 7.

Method: Use substitution. Set the equations equal to each other:

1. 2x + 1 = x + 4
2. Subtract x: x + 1 = 4
3. Subtract 1: x = 3
4. Plug x into either equation: y = 3 + 4 = 7.

For the system of equations 2x + y = 7 and x - y = 1, what are the solutions for x and y?

x = 8/3, y = 5/3.

Method: Use elimination. Add the two equations together:

1. (2x + y) + (x - y) = 7 + 1
2. 3x = 8 \implies x = 8/3
3. Substitute back: 8/3 - y = 1 \implies y = 8/3 - 3/3 = 5/3.

What does it mean if two lines never intersect in a system of equations?

It means there is no solution.

Reasoning: Parallel lines have the same slope but different y-intercepts (m1 = m2; b1 \neq b2).

What is the factored form of x^2 + 5x + 6?

(x + 2)(x + 3).

Strategy: Find two numbers that multiply to 6 and add to 5. Since 2 \times 3 = 6 and 2 + 3 = 5, the factors are (x + 2) and (x + 3).

What are the solutions for the equation x^2 - 9 = 0?

x = \pm 3.

Method: Use the Square Root Property:

1. Add 9 to both sides: x^2 = 9
2. Take the square root: x = \pm \sqrt{9} = \pm 3.

Find the vertex of the quadratic function y = x^2 - 4x + 1.

The vertex is (2, -3).

Formula: Find x using x = -\frac{b}{2a}.

1. x = -\frac{-4}{2(1)} = 2
2. Find y by plugging x into the equation: y = (2)^2 - 4(2) + 1 = 4 - 8 + 1 = -3.

Factor the expression x^2 - 16.

The factors are (x - 4)(x + 4).

Formula: Difference of Squares: a^2 - b^2 = (a - b)(a + b).
Here, a = x and b = 4.

Simplify the expression (x^2 - 4)/(x + 2).

The result is x - 2.

Method:

1. Factor the numerator: (x - 2)(x + 2)
2. Divide by the denominator: \frac{(x - 2)(x + 2)}{(x + 2)}
3. Cancel (x + 2), leaving x - 2.

Solve the equation 3^x = 9.

x = 2.

Method: Write both sides with the same base:
3^x = 3^2.
Since the bases are equal, the exponents must be equal: x = 2.

Rewrite \log_3(27) = x in exponential form.

3^x = 27.

Rule: \log_b(y) = x is equivalent to b^x = y.

What is the common difference in the sequence: 5, 9, 13, …?

The common difference is 4.

Formula: d = an - a{n-1}.
Subtract any term from the next: 9 - 5 = 4 or 13 - 9 = 4.

Write the formula for an arithmetic sequence where the first term a_1 = 3 and the common difference d = 2.

a_n = 3 + 2(n - 1).

General Formula: an = a1 + (n - 1)d.

Find the 5th term of the geometric sequence where a_1 = 2 and r = 3.

The 5th term is 162.

Formula: an = a1 \cdot r^{(n-1)}.
calc: a_5 = 2 \cdot 3^{(5-1)} = 2 \cdot 3^4 = 2 \cdot 81 = 162.

In trigonometry, what does Opposite/Hypotenuse equal?

Sine (\sin).

Mnemonic: SOH (Sine = Opposite / Hypotenuse).