For this week’s blog post, I picked the three equations of a line: slope-intercept form, point-slope form, and general form. My goal was to make myself feel more comfortable and understand that these equations so I could use them in a lot scenarios.

1. Slope-Intercept Form

Formula: y=mx + b

• m is the slope (how steep the line is).
• b is the y-intercept (where the line crosses the y -axis).

Steps:

1. Make sure the equation looks like y =mx + b.
2. The number in front of x is the slope (m).
3. The constant (b) is the y-intercept.

2. Point-Slope Form

Formula: y – y1 = M(x – x1)

• m is the slope.
(x1, y1) is a point on the line.

Steps:

1. Find the slope and a point on the line.
2. Replace m, x1, and 1 in the formula.
3. Simplify if needed.

3. General Form

Ax + By + C = 0

• A, B, and C are whole numbers.
• A should be positive, and no fractions are allowed.

Steps:

1. Move all terms to one side of the equation so it equals 0.
2. If there are fractions, multiply through to get rid of them.
3. Make A positive.

Examples

1. Slope-Intercept Form:
   A line with slope 3 and y-intercept – 2: y = 3х – 2
2. Point-Slope Form:
   A line with slope 1/2 passing through (4, -1):
   y+1=1/2(x-4）
   Simplified:
   y = 1/2x – 3

This week, I learned about parallel and perpendicular lines in math. At first, I didn’t fully understand the topic, so I decided to review it more thoroughly for my blog post. Here’s a summary of what I learned

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1. Parallel Lines

• Two lines are parallel if they never meet, no matter how far they are extended.
• Example:
  • Line 1: y=2x+3
  • Line 2: y=2x−5
  • Both lines have the same slope (m=2), so they are parallel.

2. Perpendicular Lines

• Two lines are perpendicular if they intersect at a right angle (90∘).
• For perpendicular lines, the slopes are negative reciprocals of each other.
  • If m1=a/bm_1 = a/bm1​=a/b, then m2=−b/am_2 = -b/am2​=−b/a.
  • One slope must be positive, and the other must be negative.
• Example:
  • Line 1: y=2/3x+1
  • Line 2: y=−3/2x-4
  • Since m1=2/3 m2=−3/2, the lines are perpendicular.

This week, we learned about function notation, mapping notation, and relations. I’m focusing on function notation because it’s important for understanding other math topics. Here are the steps to follow, with two simple examples:

Steps for Function Notation:

1. Know the function name:
   The name is usually a letter like f, g, or h.
2. Find the input value (x):
   The input is the number you will substitute into the function.
3. Plug in the input:
   Replace x with the given number in the equation.
4. Solve:
   Do the math to find the answer.

Example 1:

Let g(x)=4x−7
Find g(3):

Step 1: The function name is g.
Step 2: The input is x=3
Step 3: Replace x with 3:

g(3)=4(3)−7

Step 4: Solve:

g(3)=12−7=5

Answer: g(3)=5

Example 2:

Let h(x)=x2+2x−1
Find h(−2):

Step 1: The function name is h
Step 2: The input is x=−2
Step 3: Replace x with -2:

h(−2)=(−2)2+2(−2)−1

Step 4: Solve:

h(−2)=4−4−1=−1

Answer: h(−2)=−1

The domain and range in mathematics describe the inputs and outputs of a function:

1. Domain:

The domain is the set of all possible inputs ($x$-values) for which the function is defined. It excludes any values that cause division by zero, negative square roots (in real numbers), or any other undefined operations.

2. Range:

The range is the set of all possible outputs ($y$-values) that the function can produce based on its domain.

Example 1: f(x)=x^2−4

Domain: You can square any real number, so:

Domain=(−∞,∞)

Range: The smallest value of x^2 is 0 (when x=0), so the smallest output of f(x) is:

Range=[−4,∞)

Example 2: f(x)=x+3

Domain: There are no restrictions on $x$, so:

Domain=(−∞,∞)

Range: Adding 3 shifts all the outputs of $x$. Since $x$ can be any real number, the range is also:

Range=(−∞,∞)