Linear Functions & Graphs: Slope-Intercept

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✖️ 1. Slope ( $k$ ) as rate of change and the $y$ -intercept ( $b$ ) as initial value

📐 Slope and Y-Intercept Decoded

Slope $k$ tells you how much $y$ changes when $x$ increases by 1.
Positive $k$ means the line goes upward; negative $k$ means downward.
Y-intercept $b$ is where the line crosses the $y$ -axis (when $x = 0$ ).
In $y = kx + b$ , start at $b$ on the $y$ -axis, then use $k$ to find the next point.
If $k = 3$ , moving right 1 unit means moving up 3 units.

Example: $y = 2x + 5$ has slope 2 (rise 2 per 1 right) and starts at $y = 5$ .

💡 Think: $k$ is the steepness, $b$ is the launch point!

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1. Slope ( $k$ ) as rate of change and the $y$ -intercept ( $b$ ) as initial value

Slope and $y$ -Intercept in Linear Functions

A linear function has the form $y = kx + b$ , where $k$ is the slope and $b$ is the $y$ -intercept. The slope $k$ measures the rate of change of $y$ with respect to $x$ , while $b$ represents the value of $y$ when $x = 0$ .

Intuitively, $k$ tells us how steep the line is and whether it rises or falls, and $b$ gives the starting point on the vertical axis.

Core Rules:

Slope $k > 0$ : line rises from left to right
Slope $k < 0$ : line falls from left to right
Slope $k = 0$ : horizontal line (constant function)
$y$ -intercept $b$ : the point $(0, b)$ where the line crosses the $y$ -axis

These parameters completely determine the line's position and orientation in the plane.

Example: In $y = 3x + 2$ , the slope is $3$ (for each unit increase in $x$ , $y$ increases by $3$ ) and the $y$ -intercept is $2$ (the line crosses the $y$ -axis at $(0, 2)$ ).

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Given the linear function $y = -4x + 7$ , what is the slope and what does it tell us about the line?

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✖️ 2. Graphing lines using the slope-intercept method

✏️ Graphing Lines Fast

Step 1: Plot the $y$ -intercept $b$ on the $y$ -axis.
Step 2: Use slope $k = \frac{\text{rise}}{\text{run}}$ to find a second point.
Move right by the run, then up or down by the rise.
Connect the two points with a straight line and extend it.
If $k$ is a fraction like $\frac{3}{4}$ , rise 3 and run 4.

Example: For $y = -\frac{1}{2}x + 3$ , plot $(0, 3)$ , then go right 2 and down 1 to $(2, 2)$ .

💡 Remember: Start at $b$ , then follow the slope like stairs!

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2. Graphing lines using the slope-intercept method

Graphing Lines Using Slope-Intercept Method

The slope-intercept method uses the form $y = kx + b$ to graph a line by first plotting the $y$ -intercept, then using the slope to find additional points. This approach directly translates the equation's parameters into geometric steps.

Start at $(0, b)$ on the $y$ -axis, then move according to the slope $k = \frac{\text{rise}}{\text{run}}$ .

Core Rules:

Plot the $y$ -intercept: mark the point $(0, b)$
Apply the slope: from $(0, b)$ , move $k$ units vertically per $1$ unit horizontally (or use $k = \frac{m}{n}$ to move $m$ units up and $n$ units right)
Draw the line: connect the points and extend in both directions
Negative slope: move down instead of up

This method ensures accuracy and efficiency when sketching linear graphs.

Example: For $y = -2x + 1$ , plot $(0, 1)$ , then move down $2$ units and right $1$ unit to reach $(1, -1)$ ; draw the line through these points.

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If you start graphing the line $y = 3x + 2$ by plotting the y-intercept at $(0, 2)$ , what will be the y-coordinate of the next point if you move exactly $1$ unit to the right?

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✖️ 3. Determining equations of lines from graphs or two given points

🔍 Finding the Equation

From a graph: Read $b$ where the line crosses the $y$ -axis, then count rise over run for $k$ .
From two points: Calculate slope $k = \frac{y_2 - y_1}{x_2 - x_1}$ , then plug one point into $y = kx + b$ to solve for $b$ .
Always simplify $k$ to lowest terms.
Check your equation by substituting both points back in.

Example: Points $(1, 4)$ and $(3, 10)$ give $k = \frac{10 - 4}{3 - 1} = 3$ ; using $(1, 4)$ : $4 = 3(1) + b \Rightarrow b = 1$ , so $y = 3x + 1$ .

💡 Shortcut: Slope first, then plug to find $b$ !

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3. Determining equations of lines from graphs or two given points

Determining Equations from Graphs or Points

To find the equation $y = kx + b$ of a line, we need two pieces of information: the slope $k$ and the $y$ -intercept $b$ . These can be extracted from a graph or calculated from two points.

From a graph, read $b$ directly where the line crosses the $y$ -axis and compute $k$ using any two visible points. From two points $(x_1, y_1)$ and $(x_2, y_2)$ , calculate $k$ first, then solve for $b$ .

Core Rules:

Slope from two points: $k = \frac{y_2 - y_1}{x_2 - x_1}$ (where $x_2 \neq x_1$ )
Find $b$ : substitute one point and the slope into $y = kx + b$ and solve
From graph: identify intercept visually and use rise over run between grid points

This process uniquely determines the linear equation.

Example: Given points $(1, 3)$ and $(4, 9)$ , slope $k = \frac{9-3}{4-1} = 2$ ; substituting $(1, 3)$ : $3 = 2(1) + b$ gives $b = 1$ , so $y = 2x + 1$ .

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A line passes through the points $(2, 5)$ and $(4, 11)$ . Calculate the slope $k$ of this line.

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✖️ 4. Parallel and perpendicular lines: relationship between slopes

⚖️ Parallel and Perpendicular Lines

Parallel lines have identical slopes ( $k_1 = k_2$ ) but different $y$ -intercepts.
Perpendicular lines have slopes that are negative reciprocals ( $k_1 \cdot k_2 = -1$ ).
If one slope is $\frac{a}{b}$ , the perpendicular slope is $-\frac{b}{a}$ .
Horizontal lines ( $k = 0$ ) are perpendicular to vertical lines (undefined slope).

Example: $y = 2x + 1$ is parallel to $y = 2x - 5$ ; perpendicular to $y = -\frac{1}{2}x + 3$ since $2 \cdot (-\frac{1}{2}) = -1$ .

💡 Visual: Parallel = same tilt; perpendicular = flip and negate!

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4. Parallel and perpendicular lines: relationship between slopes

Parallel and Perpendicular Lines

Two non-vertical lines are parallel if they have equal slopes, and perpendicular if the product of their slopes equals $-1$ . These slope relationships determine the geometric configuration of lines in the plane.

Parallel lines never intersect because they rise at the same rate; perpendicular lines intersect at right angles.

Core Rules:

Parallel lines: $k_1 = k_2$ (same slope, different $y$ -intercepts)
Perpendicular lines: $k_1 \cdot k_2 = -1$ , equivalently $k_2 = -\frac{1}{k_1}$ (negative reciprocal)
Vertical and horizontal: a vertical line (undefined slope) is perpendicular to any horizontal line (slope $0$ )
Exception: vertical lines cannot be expressed in slope-intercept form

These conditions are both necessary and sufficient for the respective geometric relationships.

Example: Lines $y = 3x + 1$ and $y = 3x - 5$ are parallel (both have slope $3$ ); lines $y = 2x + 1$ and $y = -\frac{1}{2}x + 4$ are perpendicular since $2 \cdot (-\frac{1}{2}) = -1$ .

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A line is given by the equation $y = 6x - 2$ . What is the slope of any line that is parallel to this line?

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✖️ 5. Applications: Modeling constant speed motion in physics or simple tax/utility bills in economics

🌍 Real-World Linear Models

Constant speed: Distance $d = vt + d_0$ where $v$ is speed (slope) and $d_0$ is starting position.
Utility bills: Cost $C = rt + f$ where $r$ is rate per unit (slope) and $f$ is fixed fee.
Tax calculations: Total $T = px + b$ where $p$ is tax rate and $b$ is base amount.
The slope always represents the rate of change per unit.
The $y$ -intercept is the starting value when input is zero.

Example: A taxi charges 3 dollars per km plus 5 dollars flat fee: $C = 3x + 5$ where $x$ is km traveled.

💡 Key: Slope = rate, intercept = starting point in any scenario!

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5. Applications: Modeling constant speed motion in physics or simple tax/utility bills in economics

Applications of Linear Functions

Linear functions model situations where one quantity changes at a constant rate relative to another. In physics, constant speed motion follows $d = vt + d_0$ (distance equals speed times time plus initial position). In economics, utility bills often have the form $C = rq + f$ (total cost equals rate per unit times quantity plus fixed fee).

The slope represents the constant rate (speed, price per unit), and the $y$ -intercept represents the initial condition (starting position, base fee).

Core Rules:

Physics motion: $k$ is velocity (meters per second), $b$ is initial position
Economics billing: $k$ is unit rate (dollars per kilowatt-hour), $b$ is fixed charge
Interpretation: slope gives marginal change, intercept gives baseline value
Domain restrictions: often $x \geq 0$ in real applications

Linear models provide simple yet powerful tools for prediction and analysis.

Example: A taxi charges 3 dollars per kilometer plus a 5 dollar base fare; the cost function is $C = 3d + 5$ , where $d$ is distance in kilometers.

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A utility company charges a fixed fee of 15 dollars per month, plus 0.20 dollars for each kilowatt-hour used. If $q$ represents the kilowatt-hours used, what is the value of the $y$ -intercept in the linear function modeling the total cost?

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Linear function and its graph ( $y = kx + b$ )

✖️ 1. Slope ( $k$ ) as rate of change and the $y$ -intercept ( $b$ ) as initial value

📐 Slope and Y-Intercept Decoded

1. Slope ( $k$ ) as rate of change and the $y$ -intercept ( $b$ ) as initial value

Slope and $y$ -Intercept in Linear Functions

✖️ 2. Graphing lines using the slope-intercept method

✏️ Graphing Lines Fast

2. Graphing lines using the slope-intercept method

Graphing Lines Using Slope-Intercept Method

✖️ 3. Determining equations of lines from graphs or two given points

🔍 Finding the Equation

3. Determining equations of lines from graphs or two given points

Determining Equations from Graphs or Points

✖️ 4. Parallel and perpendicular lines: relationship between slopes

⚖️ Parallel and Perpendicular Lines

4. Parallel and perpendicular lines: relationship between slopes

Parallel and Perpendicular Lines

✖️ 5. Applications: Modeling constant speed motion in physics or simple tax/utility bills in economics

🌍 Real-World Linear Models

5. Applications: Modeling constant speed motion in physics or simple tax/utility bills in economics

Applications of Linear Functions

AWAITING_CONFIRMATION

✖️ 1. Slope (kkk) as rate of change and the yyy-intercept (bbb) as initial value

📐 Slope and Y-Intercept Decoded

1. Slope (kkk) as rate of change and the yyy-intercept (bbb) as initial value

Slope and yyy-Intercept in Linear Functions

✖️ 2. Graphing lines using the slope-intercept method

✏️ Graphing Lines Fast

2. Graphing lines using the slope-intercept method

Graphing Lines Using Slope-Intercept Method

✖️ 3. Determining equations of lines from graphs or two given points

🔍 Finding the Equation

3. Determining equations of lines from graphs or two given points

Determining Equations from Graphs or Points

✖️ 4. Parallel and perpendicular lines: relationship between slopes

⚖️ Parallel and Perpendicular Lines

4. Parallel and perpendicular lines: relationship between slopes

Parallel and Perpendicular Lines

✖️ 5. Applications: Modeling constant speed motion in physics or simple tax/utility bills in economics

🌍 Real-World Linear Models

5. Applications: Modeling constant speed motion in physics or simple tax/utility bills in economics

Applications of Linear Functions

AWAITING_CONFIRMATION

✖️ 1. Slope ( $k$ ) as rate of change and the $y$ -intercept ( $b$ ) as initial value

1. Slope ( $k$ ) as rate of change and the $y$ -intercept ( $b$ ) as initial value

Slope and $y$ -Intercept in Linear Functions