Best Digital SAT Math Prep: "Linear Functions"

Frontier Lesson: "Linear Functions" for the Digital SAT

"Linear Functions" is a key topic in the "Algebra" content domain of the SAT! Mastering linear functions is not only essential for the test but also for building a strong foundation in math.

On the Digital SAT, you'll encounter questions about linear functions in multiple-choice and student-produced response (grid-in) formats. The good news? These questions tend to range from easy to medium difficulty, and even the "hard" ones are manageable once you understand the core concepts and tricks.

🎯 Here's what you'll often see:

Solving for variables or constants in given linear functions
Finding slope, x-intercept, or y-intercept from linear equations
Writing linear equations based on word problems or graphs
Applying linear relationships to solve real-world problems (e.g., rate, cost, growth)

Definition and Basic Forms of "Linear Functions"

➢ Two Equivalent Forms of "Linear Functions"

Slope-Intercept Form: $y = k x + b$ (most common)
- $k$ : Slope (rate of change), $b$ : y-intercept (y-coordinate of the intersection with the y-axis).
- Example: $y = - 2 x + 1$ has a slope $k = - 2$ and y-intercept $b = 1$ .
Standard Form: $A x + B y = C$ ( $A$ and $B$ not both zero)
- Can be converted to slope-intercept form: $y = - \frac{A}{B} x + \frac{C}{B}$ (if $B \neq = 0$ ).
- Example: $3 x + 2 y = 6$ → $y = - \frac{3}{2} x + 3$ .

➢ Key Parameters of "Linear Functions"

(1) Slope (k)

Geometric Meaning:

$k = \frac{vertical change}{horizontal change} = \frac{Δ y}{Δ x}$

Example: A slope $k = \frac{1}{2}$ means the line rises 1 unit for every 2 units moved to the right.

Classification and Graph Characteristics:

Slope $k$	Function Type	Graph Direction	Passes Through Quadrants
$k > 0$	Increasing	Upward to the right	Always passes through Quadrants I and III
$k < 0$	Decreasing	Downward to the right	Always passes through Quadrants II and IV
$k = 0$	Constant	Horizontal line	Parallel to x-axis ( $y = b$ )

(2) Intercepts

y-intercept (b):
- Intersection with the y-axis at $(0, b)$ .
- Example: $y = - 3 x - 4$ has a y-intercept of $- 4$ .
x-intercept:
- Set $y = 0$ , solve for $x = - \frac{b}{k}$ (slope-intercept form) or $x = \frac{C}{A}$ (standard form).
- Example: $y = 6 x + 2$ has an x-intercept at $x = - 2$ .

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Graphs and Properties of "Linear Functions"

➢ Graphs and Key Characteristics

Graphing Methods:
- Slope-Intercept Form: Plot the point $(0, b)$ and another point (e.g., for $x = 1$ , calculate $y = k + b$ ).
- Standard Form: Find x- and y-intercepts, then connect them (e.g., $2 x - y = 4$ has intercepts at $(2, 0)$ and $(0, - 4)$ ).
Symmetry:
- If $b = 0$ (proportional function), the graph is symmetric about the origin (e.g., $y = \frac{1}{4} x$ ).

➢ Special Linear Functions

Type	Equation	Properties	Graph (below)
Proportional Function	$y = k x$	Passes through the origin; slope determines direction (you can review the previous table and graph: $k > 0$ for Quadrants I/III; $k < 0$ for II/IV)	Diagonal line through origin
Constant Function	$y = b$	Horizontal line; no slope ( $k = 0$ )	Parallel to x-axis
Vertical Line	$x = a$	Not a function (undefined slope)	Perpendicular to x-axis

➢ Parallel and Perpendicular Lines

1. Parallel Lines

Condition: Same slope ( $k_{1} = k_{2}$ ).
Example:
- $y = 2 x + 1$ and $y = 2 x - 3$ are parallel.

2. Perpendicular Lines

Condition: Slopes satisfy $k_{1} \times k_{2} = - 1$ (e.g., one is $\frac{1}{3}$ , the other is $- 3$ ).
Example:
- $y = \frac{1}{3} x - 2$ and $y = - 3 x - 10$ are perpendicular.
Special Cases:
- A horizontal line ( $y = b$ ) is perpendicular to any vertical line ( $x = a$ ).

What Information Is Needed to Write the Equation of A "Linear Function"?

In real Digital SAT, you will encounter two situations:

➢ Given: Slope $k$ and a Point $(x_{0}, y_{0})$

Method: Substitute into slope-intercept form $y = k x + b$ and solve for $b$ .

Example Steps:

Given slope $k = 3$ and point $(1, 5)$ :

Substitute into slope-intercept form:
$5 = 3 \times 1 + b$ → $b = 2$
Final equation:
$y = 3 x + 2$

Alternative Point-Slope Method (additional note):
$y - y_{0} = k (x - x_{0})$ → $y - 5 = 3 (x - 1)$ → $y = 3 x + 2$

➢ Given: Two Points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$

Example Steps:

Given points $(2, 3)$ and $(4, 7)$ :

Calculate slope:
$k = \frac{7 - 3}{4 - 2} = 2$
Thus, we can have a slope-intercept form $y = 2 x + b$
Choose one point (e.g., $(2, 3)$ ) and substitute into the slope-intercept form:
$3 = 2 \times 2 + b$ → $b = - 1$
Final equation:
$y = 2 x - 1$

Common SAT "Linear Functions" Question Types

Example 1. Solving for Variables or Constants in Linear Functions

These questions give you a linear function (e.g., $f (x) = m x + b$ ) and ask you to find a missing variable or constant, such as:

The function $f$ is defined by $f (x) = 31 x + 4$ . What is the value of $f (4)$ ?
The function $f$ is defined by $f (x) = 9 x$ . For what value of $x$ does $f (x) = 81$ ?
If $f (x) = x + 3$ and $g (x) = 7 x$ , what is the value of $5 f (2) - g (2)$ ?
For the linear function $f (x) = - 2 x + b$ , $b$ is a constant and $f (- 2) = 8$ . What is the value of $b$ ?

What to Do:
Plug the given values into the equation and solve for the unknown. These problems are usually straightforward—just be careful not to mix up variables (like $x$ ) and functions (like $f (x)$ ) while calculating. Follow basic algebra steps (add/subtract/multiply/divide) to isolate the variable, and you'll get the answer.

Example:
"If $f (x) = 2 x - 5$ and $f (a) = 7$ , what is the value of $a$ ?"
→ Replace $f (x)$ with $7$ : $7 = 2 a - 5$
→ Solve: $2 a = 12$ , so $a = 6$ .

📣 Be Careful:

Stay organized: Label each term clearly (e.g., slope, constant, input/output).
Double-check signs (+/−) when moving terms across the equation.

Example 2. Finding Slope, x-intercept, or y-intercept

You may need to identify key features of a linear equation, such as:

Slope (m): The rate of change (e.g., in $y = m x + b$ , $m$ is the slope).
y-intercept (b): The point where the line crosses the y-axis (at $x = 0$ ).
x-intercept: The point where the line crosses the x-axis (found by setting $y = 0$ ).

Examples:

The function $f$ is defined by $f (x) = - \frac{11}{12} x - 77$ . What is the y-intercept of the graph of $y = f (x)$ in the $x y$ -plane?
The function $f (x)$ is defined as $10$ more than $3$ times a number $x$ . If $y = f (x)$ is graphed in the $x y$ -plane, what is the best interpretation of the $x$ -intercept?

What to Do:

Understand the what does "slope/ $x$ -intercept/ $y$ -intercept" mean.
Rewrite the equation in slope-intercept form ( $y = m x + b$ ) if needed.
Identify the intercepts by setting one variable to zero and solving for the other.

Example 3. Defining Linear Equations from Word Problems or Graphs

Some questions present a scenario or a graph and ask you to write the corresponding equation.

Sample word problems:
- A taxi charges a $3 base fee plus $2 per mile. Which equation can be defined for the total cost $C$ in terms of miles $m$ ?
- In the $x y$ -plane, the graph of the linear function $f$ contains the point $(0, 4)$ and $(8, 23)$ . Which equation defines $f$ , where $y = f (x)$ ?
Sample graph questions:

The graph of the linear function $y = f (x) + 15$ is shown. If $c$ and $d$ are positive constants, which equation could define $f$ ?

What to Do:

For giving two points or slope + one point:
- If given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , first calculate the slope ( $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ ).
- Then, plug the slope and one point into point-slope form (or plug into the slope-intercept form): $y - y_{1} = m (x - x_{1})$ .
- Simplify to slope-intercept form ( $y = m x + b$ ) if needed.
For word problems:
- Identify the variables (the "per" quantity, like cost per mile) and the constant (fixed value, like a base fee).
- Set up the equation: Total = (Rate × Variable) + Initial Value.
- Example: "A gym charges $30/month plus a $50 sign-up fee" → $C = 30 m + 50$ .
Verify your equation:
- Plug in a known point to check if it satisfies your equation.
- For word problems, test a simple case (e.g., "After 0 months, is the cost equal to the initial fee?").

Example 4. Applying Linear Relationships to Real-life Situations Word Problems

1. What These Questions Ask You to Do:

Solve for a specific value (e.g., "After 5 hours, how far has the car traveled?")
Write a linear equation modeling the situation (e.g., "Which equation defines the total cost after buying $n$ tickets?")
Interpret a variable/constant (e.g., "..., which of the following is the best interpretation of 55 in this context?")

These problems test your ability to translate real-world scenarios into math and extract meaningful information.

2. Common Real-Life Situations

You'll often encounter linear relationships in these contexts:

Speed/Distance/Time:
Cost/Revenue/Profit:
Temperature Changes:
Population/Growth:
Money Savings/Earnings:

How to Approach These Problems:

Identify the variables and constants—what changes ("per hour", "per item") vs. what stays fixed (initial value, base fee).
Set up the equation in $y = m x + b$ form, where:
- $m$ = rate of change (slope)
- $b$ = starting value (y-intercept)
Solve or interpret based on the question's goal.
Double-check units (hours vs. minutes, dollars vs. cents) to avoid mistakes.

Example Question:
A taxi charges a $5 base fare plus $2 per mile. If you paid $21, how many miles did you ride?

Rate of change (slope): $2$ per mile
Constant (y-intercept): $5$ is a fixed cost
Equation: $Cost = 2 x + 5$
Solve: $21 = 2 x + 5$ → $x = 8$ miles

*Common Trap: Forgetting the base fee and solving $21 = 2 x$ → Wrong answer: $10.5$ miles!

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Cheatsheets During the Actual Digital SAT

🚀 Quick Analysis Using Graphs

Given: $f (x) = k x + b$

➥ Determine the sign of the slope by observing the quadrants the line passes through

Positive slope: $k > 0$ : The line goes from bottom-left to top-right, passing through the first and third quadrants
Negative slope: $k < 0$ : The line goes from top-left to bottom-right, passing through the second and fourth quadrants

➥ Determine the sign of the y-intercept( $b$ ) by observing where the line crosses the y-axis

Positive y-intercept: $b > 0$ : The intersection point is above the x-axis
Negative y-intercept: $b < 0$ : The intersection point is below the x-axis
No y-intercept: $b = 0$ : The intersection point passes through the origin $(0, 0)$

💡 This method helps you quickly eliminate incorrect answer choices. Sometimes you don't even need to perform actual calculations - you can select the correct answer just by analyzing the graph.

Example:

Explanation:

Based on the graph, we can first observe that the function passes through the second and fourth quadrants, which means the slope is negative. At this point, we can immediately eliminate options A and D (both have a slope of 1/3). Next, we only need to choose between B and C.
Then, looking at where the line intersects the y-axis, we see it's below the x-axis-that is, on the negative half of the y-axis. This means the y-intercept is negative, so option B is also eliminated (its y-intercept is +6).
Thus, the correct answer is C.

In this process, we didn't pick points on the graph and calculate the function's equation to solve it. We could determine the answer just by "looking at the graph directly"!

🚀 Quick Calculation of Slope

Master the slope formula for two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ :
$k = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Do not swap the numerator and denominator.
If the SAT problem provides two intercept points (e.g., $(3, 0)$ and $(0, 4)$ ), the slope is:
$k = - \frac{y -intercept}{x -intercept} = - \frac{4}{3}$ .

🚀 Intercept Shortcut Method

Memorize these conclusions to save time during exams:
- x-intercept $= - \frac{b}{k}$ (derived by setting $y = 0$ in $0 = k x + b$ )
- y-intercept $= b$ (obtained directly by setting $x = 0$ )
Note: If the equation is in standard form $A x + B y = C$ , then:
- x-intercept $= \frac{C}{A}$
- y-intercept $= \frac{C}{B}$ (no rearrangement needed).

🚀 Two-Point Form to Slope-Intercept Form

Given two points, first calculate the slope $k$ , then use the point-slope form:
$y - y_{1} = k (x - x_{1})$
Finally, rearrange it to:
$y = k x + b$ .

🚀 Quick Graph Reading

y-intercept: Observe the y-coordinate where the line crosses the y-axis.
Slope: Count the grid units for $\frac{rise}{run}$ (e.g., a point moves right 2 units and up 1 unit to return to the line → $k = \frac{1}{2}$ ).

🚀 Keyword Translation in Word Problems

Terms like "Initial value", "base fee", "fixed cost", etc. → y-intercept $b$ .
Terms like "Rate of change", ".. per ..", etc. → Slope $k$ .

🚀 Extreme Value Verification

Test $x = 0$ or $x = 1$ to validate the equation's reasonability.

🚀 Reverse Substitution for Multiple-Choice

For complex functions (e.g., fractions), if the question asks for $f (3)$ , substitute the given options backward to check which fits the function.

🚀 Simplify Equations

First, check if the function can be simplified (e.g., $2 g (x) = 4 f (x) - 8$ simplifies to $g (x) = 2 f (x) - 4$ ). This makes analysis clearer and reduces calculation difficulty.

Quick Practice: Test Your Skills!

Question 1

In the $x y$ -plane, the graph of the linear function $f (x)$ passes through the points $(2, 5)$ and $(- 8, - 17)$ . Which of the following defines $f (x)$ ?

A) $y = \frac{3}{5} x + \frac{11}{5}$

B) $y = \frac{11}{5} x + \frac{3}{5}$

C) $y = - \frac{11}{5} x + \frac{3}{5}$

D) $y = \frac{11}{5} x - \frac{3}{5}$

Explanation:

Step 1: Find the slope using the formula $k = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ :
$k = \frac{5 - ( - 17 )}{2 - ( - 8 )} = \frac{22}{10} = \frac{11}{5}$

Thus, Option A and option C can be eliminated.

Step 2: Use the point-slope form with one of the points, let's use $(2, 5)$ :

$y - 5 = \frac{11}{5} (x - 2)$

$y - 5 = \frac{11}{5} x - \frac{22}{5}$

$y = \frac{11}{5} x - \frac{22}{5} + 5$

$y = \frac{11}{5} x - \frac{22}{5} + \frac{25}{5}$

$y = \frac{11}{5} x + \frac{3}{5}$

Or, you can substitute one point, for example $(2, 5)$ , to either option B or option D to check if the result is correct. Let's test $x = 2$ in option B:

$y = \frac{11}{5} \times 2 + \frac{3}{5} = \frac{22}{5} + \frac{3}{5} = \frac{25}{5} = 5$ , which matches the given point $(2, 5)$ .

Finally, the correct answer is B

Question 2

A worker repairs wooden boxes and mows the lawn. The formula $T (x) = 15 x + 70$ relates the total time $T$ , in minutes, it takes to repair $x$ wooden boxes and mow the lawn. Which of the following describes the meaning of the $70$ in this context?

A) The increase in time, in minutes, it takes the worker to repair a wooden box.

B) The time, in minutes it takes the worker to mow the lawn.

C) The increase in time, in minutes, it takes the worker to repair a wooden box and mow the lawn.

D) The total time, in minutes, it takes the worker to repair the wooden boxes.

Explanation:

The equation $T (x) = 15 x + 70$ represents a linear function. In this context:
- $x$ is a variable and represents the number of wooden boxes repaired.
- $15$ is rate of change and represents the time spent repairing each box.
- $70$ is a fixed value. It also represents the y-intercept of a linear function, which is the value of $T (x)$ when $x = 0$ (meaning no boxes are repaired). This constant time would be related to something other than repairing boxes, and the only activity mentioned is mowing the lawn.

Thus, The correct answer is B.

Question 3

Explanation:

Step 1: Determine the sign of the slope
From the graph, we can notice that the line passes through the second and fourth quadrants, which means the slope must be negative. So we can quickly eliminate option A and option B (positive slope $\frac{8}{11}$ and $\frac{11}{8}$ )
Step 2: We can see that option C and option D have the same $y$ -intercept: $- \frac{3}{4}$ , so we need to further determine the value of slope. Let $k$ = slope and the equation can be written as $f (x) = k x - \frac{3}{4}$
Step 3: Plug a point into the equation.
From the graph, we can see the point $(- 2, 2)$ is on the line. Let's solve for $k$ : $2 = - 2 k - \frac{3}{4}$
- Multiply both sides by 4 to remove the denominator: $4 \times (- 2 k - \frac{3}{4}) = 2 \times 4 ⟹ - 8 k - 3 = 8 ⟹ - 8 k = 11 ⟹ k = - \frac{11}{8})$
Thus, the function $f$ is $f (x) = - \frac{11}{8} x - \frac{3}{4}$ which matches option D

Question 4

A car rental company charges a flat fee of $ $25$ plus $ $0.15$ per mile driven. If Jamal paid $ $70$ for his rental, how many miles did he drive?

A) $300$ miles
B) $450$ miles
C) $250$ miles
D) $350$ miles

Explanation:

Step 1: Write the linear function for the cost. Let $x$ = number of miles driven and $C (x)$ = total cost.
- Rate of change (slope): $0.15$ per mile driven
- Constant (y-intercept): $25$ is a fixed cost
- Equation: $C (x) = 0.15 x + 25$
Step 2: Set up an equation using the given information: $70 = 0.15 x + 25$
Step 3: Solve for $x$
$70 - 25 = 0.15 x ⟹ 45 = 0.15 x ⟹ x = \frac{45}{0.15} = 300$

Thus, Jamal drove 300 miles, which matches option A.

Question 5

$x$	$g (x)$
1	29
2	24
3	19
4	14

For the linear function $g$ , the table shows four values of $x$ and their corresponding values of $g (x)$ . The function can be written as $g (x) = m x + b$ , where $m$ and $b$ are constants. What is the value of $b$ ?

Explanation:

Step 1: Determine the slope ( $m$ )
The slope of a linear function can be calculated using any two points from the table. Let's use the first two points $(1, 29)$ and $(2, 24)$ :
$m = \frac{g ( x _{2} ) - g ( x _{1} )}{x _{2} - x _{1}} = \frac{24 - 29}{2 - 1} = \frac{- 5}{1} = - 5$
So, the slope $m$ is $- 5$ .
Step 2: Find the y-intercept ( $b$ )
Now that we have $m = - 5$ , we can plug one of the points into the equation $g (x) = m x + b$ to solve for $b$ . Let's use the first point $(1, 29)$ :
$29 = (- 5) (1) + b ⟹ 29 = - 5 + b ⟹ b = 29 + 5 = 34$
Thus,the value of $b$ is $34$

Your Turn! Realistic "Linear Functions" Questions for DSAT Success

Question 1

Difficulty level: Easy

$f (x) = 5 x + b$

For the linear function $f, b$ is a constant and f(7) = 35. What is the value of b?

A). $0$

B). $1$

C). $5$

D). $7$

Question 2

Difficulty level: Medium

The function $f (x)$ is defined as $17$ more than $3$ times a number $x$ . If $y = f (x)$ is graphed in the $x y$ -plane, what is the best interpretation of the $x$ -intercept?

A). When $f (x) = 0$ , the number is $- \frac{17}{3}$ .

B). When the number is $0$ , $f (x) = 17$ .

C). For each increase of $1$ in the value of the number, $f (x)$ increases by $3$ .

D). The value of $f (x)$ increases by $1$ for each increase of $3$ in the value of the number.

Question 3

Difficulty level: Hard

The graph of the linear function $y = f (x) + 11$ is shown. If $c$ and $d$ are positive constants, which equation could define $f$ ?

A). $f (x) = - d - c x$

B). $f (x) = d + c x$

C). $f (x) = - d + c x$

D). $f (x) = d - c x$

"Linear Functions" Learning Checklist

🔘 Know two equivalent forms of a linear function:
- Slope-Intercept Form: $y = k x + b$ most common
- Standard Form: $A x + B y = C$ ( $A$ and $B$ not both zero)
🔘 Be able to correctly identify the slope, $y$ -intercept, $x$ -intercept from a linear equation.
🔘 Can write and convert between different forms of linear equations (slope-intercept, point-slope, standard form).
🔘 Understand special linear functions:
- Proportional Function
- Constant Function
- Vertical Line(Not a function)
🔘 Know how to calculate the slope $k$ between two points: $k = \frac{vertical change}{horizontal change} = \frac{Δ y}{Δ x}$
🔘 Be able to write the equation of a line given various information:
- slope and y-intercept
- slope and a point
- two points
🔘 Be familiar with the common "Linear Functions" question types in real Digital SAT:
- Solve for variables or constants
- Find slope, $x$ -intercept, or $y$ -intercept
- Define a linear equation from words or a graph
- Apply linear relationships to real-life situations
🔘 Master practical strategies to solve questions faster:
- Determine the sign of the slope by observing the quadrants the line passes through
- Determine the y-intercept by observing where the line crosses the y-axis
- Calculate the value of slope in a more efficient way
- Memorize conclusions to slove for the $x$ -intercept and $y$ -intercept
- Use point-slope form to get the needed slope-intercept form of a linear function
- Count the grid units to obtain the slope
- Translate the keywords in proper way in word questions
- Test the equation quickly by using extreme values
- Substitute the given options backward to check if they fit the given function.
- Always simplify the equation to reduce calculation difficulty.

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Best Digital SAT Math Prep: "Linear Functions"

Frontier Lesson: "Linear Functions" for the Digital SAT

Definition and Basic Forms of "Linear Functions"

➢ Two Equivalent Forms of "Linear Functions"

➢ Key Parameters of "Linear Functions"

(1) Slope (k)

(2) Intercepts

Free Full Length SAT Tests withOfficial-StyleQuestions

Graphs and Properties of "Linear Functions"

➢ Graphs and Key Characteristics

➢ Special Linear Functions

➢ Parallel and Perpendicular Lines

1. Parallel Lines

2. Perpendicular Lines

What Information Is Needed to Write the Equation of A "Linear Function"?

➢ Given: Slope k and a Point (x0​,y0​)

➢ Given: Two Points (x1​,y1​) and (x2​,y2​)

Common SAT "Linear Functions" Question Types

Example 1. Solving for Variables or Constants in Linear Functions

Example 2. Finding Slope, x-intercept, or y-intercept

Example 3. Defining Linear Equations from Word Problems or Graphs

Example 4. Applying Linear Relationships to Real-life Situations Word Problems

1. What These Questions Ask You to Do:

2. Common Real-Life Situations

How to Approach These Problems:

Free Full Length SAT Tests withOfficial-StyleQuestions

Cheatsheets During the Actual Digital SAT

🚀 Quick Analysis Using Graphs

🚀 Quick Calculation of Slope

🚀 Intercept Shortcut Method

🚀 Two-Point Form to Slope-Intercept Form

🚀 Quick Graph Reading

🚀 Keyword Translation in Word Problems

🚀 Extreme Value Verification

🚀 Reverse Substitution for Multiple-Choice

🚀 Simplify Equations

Quick Practice: Test Your Skills!

Question 1

Question 2

Question 3

Question 4

Question 5

Your Turn! Realistic "Linear Functions" Questions for DSAT Success

Question 1

Question 2

Question 3

"Linear Functions" Learning Checklist

Essential SAT Prep Tools

Personalized Study Planner

Expert-Curated Question Bank

Smart Flashcards

Score Calculator

SAT Skills Lessons

Full-Length Practice Tests

Pro Tip

➢ Given: Slope $k$ and a Point $(x_{0}, y_{0})$

➢ Given: Two Points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$