Worked Examples: SAT Strategy
Example 1: Easy (Finding the Equation from a Point and Slope)
Question: Line $r$ in the $xy$-plane has a slope of $4$ and passes through the point $(0, 6)$. Which equation defines line $r$?
SAT Strategy: Recognize definitions immediately. The point $(0, 6)$ is the $y$-intercept because its $x$-coordinate is $0$.
- The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
- Substitute $m = 4$ and $b = 6$ directly into the equation.
- Answer: $y = 4x + 6$.
Example 2: Medium (Interpreting Linear Models)
Question: The relationship between two variables, $x$ and $y$, is linear. For every increase in the value of $x$ by 1, the value of $y$ increases by 8. When the value of $x$ is 2, the value of $y$ is 18. Which equation represents this relationship?
SAT Strategy: Translate word problems into mathematical components. "Increase in $y$ for every increase in $x$ by 1" is the exact definition of slope ($m$).
- Identify the slope: $m = 8$. The equation is $y = 8x + b$.
- Use the given coordinate point $(2, 18)$ to find $b$.
- Substitute: $18 = 8(2) + b \implies 18 = 16 + b \implies b = 2$.
- Answer: $y = 8x + 2$.
Example 3: Hard (Perpendicular Lines)
Question: Line $k$ is defined by $y = 7x + \frac{1}{8}$. Line $j$ is perpendicular to line $k$ in the $xy$-plane. What is the slope of line $j$?
SAT Strategy: AI SL students are not given the perpendicular slope formula on the SAT reference sheet. You must memorize that perpendicular lines have
negative reciprocal slopes ($m_1 \times m_2 = -1$).
- Identify the slope of line $k$. The equation is in $y=mx+b$ form, so $m_k = 7$.
- To find the perpendicular slope, flip the fraction and change the sign.
- $7$ becomes $\frac{1}{7}$, and positive becomes negative.
- Answer: $-\frac{1}{7}$.
Practice Questions
Level 1: Easy (Foundational SAT)
- For the linear function $h$, $h(x) = x + b$, where $b$ is a constant. If $h(0) = 45$, what is the value of $b$? (Grid-in)
- A line in the $xy$-plane has a slope of $\frac{1}{9}$ and passes through the point $(0, 14)$. Which equation represents this line?
- A) $y = -\frac{1}{9}x - 14$
- B) $y = -\frac{1}{9}x + 14$
- C) $y = \frac{1}{9}x - 14$
- D) $y = \frac{1}{9}x + 14$
- Line $r$ in the $xy$-plane has a slope of $4$ and passes through the point $(0, 6)$. Which equation defines line $r$?
- A) $y = 4x + 6$
- B) $y = 4x - 6$
- C) $y = 6x + 4$
- D) $y = -6x + 4$
- The function $f$ is defined by $f(x) = 2x + 3$. The graph of $y = f(x)$ in the $xy$-plane is parallel to line $j$. What is the slope of line $j$? (Grid-in)
- The table below shows three values of $x$ and their corresponding values of $y$. There is a linear relationship between $x$ and $y$.
{|c|c|}
$x$ & $y$
$0$ & $18$
$1$ & $13$
$2$ & $8$
Which of the following equations represents this relationship?
- A) $y = 18x + 13$
- B) $y = -5x + 18$
- C) $y = -5x + 13$
- D) $y = 18x - 5$
Level 2: Medium (Standard SAT)
- The table shows selected values from the linear function $f$.
{|c|c|}
$x$ & $f(x)$
$0$ & $29$
$1$ & $32$
Which equation defines $f(x)$?
- A) $f(x) = 29x + 32$
- B) $f(x) = 32x + 29$
- C) $f(x) = 3x + 29$
- D) $f(x) = 3x + 32$
- The relationship between two variables, $x$ and $y$, is linear. For every increase in the value of $x$ by 1, the value of $y$ increases by 8. When the value of $x$ is 2, the value of $y$ is 18. Which equation represents this relationship?
- A) $y = 2x + 18$
- B) $y = 8x + 2$
- C) $y = 8x + 18$
- D) $y = 2x + 8$
- The function $f$ is defined by $f(x) = 7x - 84$. What is the $x$-intercept of the graph of $y = f(x)$ in the $xy$-plane?
- A) $(-12, 0)$
- B) $(0, -84)$
- C) $(12, 0)$
- D) $(84, 0)$
- For the linear function $f$, $f(x) = 4x + b$, where $b$ is a constant. If $f(7) = 28$, what is the value of $b$? (Grid-in)
- The table shows the linear relationship between the number of cars, $c$, on a commuter train and the maximum number of passengers and crew, $p$, that the train can carry.
{|c|c|}
Number of cars ($c$) & Maximum passengers and crew ($p$)
$3$ & $174$
$5$ & $284$
$10$ & $559$
Which equation represents the linear relationship between $c$ and $p$?
- A) $p = 55c + 9$
- B) $p = 55c - 9$
- C) $c = 55p + 9$
- D) $p = 110c + 9$
Level 3: Hard (Advanced SAT)
- Line $k$ is defined by $y = 7x + \frac{1}{8}$. Line $j$ is perpendicular to line $k$ in the $xy$-plane. What is the slope of line $j$?
- A) $-7$
- B) $-\frac{1}{7}$
- C) $\frac{1}{7}$
- D) $7$
- What is the equation of the line that passes through the points $(0, -8)$ and $(-8, 0)$ in the $xy$-plane?
- A) $y = -x - 8$
- B) $y = x - 8$
- C) $y = -x + 8$
- D) $y = x + 8$
- A linear model predicts the number of active projects a company was working on $x$ months after the end of November 2012. If the model is defined by the equation $y = -x + 5$, what is the predicted number of active projects the company was working on at the exact end of November 2012? (Grid-in)
- The linear function $f$ is defined by $f(x) = -2x + c$, where $c$ is a constant. The graph of $y = f(x)$ in the $xy$-plane intersects the $x$-axis at $(4, 0)$. What is the $y$-coordinate of the $y$-intercept of the graph of $f$? (Grid-in)
- A line in the $xy$-plane passes through the points $(2, 5)$ and $(4, 9)$. What is the $y$-coordinate of the $y$-intercept of this line? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- Function $f$ is a linear function. The graph of $y = g(x)$ in the $xy$-plane is the result of shifting the graph of $y = f(x)$ down 3 units. If $g(x) = 4x - 5$, what is the $x$-coordinate of the $x$-intercept of the graph of $y = f(x)$? (Grid-in)
- Line $p$ has the equation $ax + by = c$, where $a$, $b$, and $c$ are positive constants. What is the slope of a line that is perpendicular to line $p$ in the $xy$-plane?
- A) $-\frac{a}{b}$
- B) $\frac{a}{b}$
- C) $-\frac{b}{a}$
- D) $\frac{b}{a}$
- For a linear function $h$, $h(3) = 14$ and $h(7) = 34$. What is the value of $h(10)$? (Grid-in)
- The graph of linear function $f$ in the $xy$-plane passes through the origin and the point $(c, d)$, where $c \neq 0$. The graph of linear function $g$ passes through the origin and the point $(c, -d)$. If $f(5) = 10$, what is the value of $g(5)$? (Grid-in)
- Line $L$ has the equation $y = 3x + b$, where $b$ is a constant. Line $M$ is perpendicular to line $L$ and passes through the point $(0, 4)$. If the two lines intersect at the point $(6, d)$, what is the value of $b$? (Grid-in)
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