Coordinate Point Patterns - SHSAT Math

1. On the coordinate plane shown, points are plotted at $(1,3)$, $(2,6)$, $(4,12)$, and $(5,15)$, but the point for $x=3$ is missing. If the pattern continues linearly, and the sum of ALL six $y$-coordinates (including $x=3$ and $x=6$) equals what value?
   Answer: $63$ The rule is $y=3x$. For $x=1,2,3,4,5,6$: $y=3,6,9,12,15,18$. Sum = $63$. Choice B omits $x=3$: $54$. Choice C omits $x=6$. Choice D sums only $x=1$ through $5$.
2. In the coordinate plane, point $A$ is at $(3, -2)$ and point $B$ is at $(7, 6)$. Point $C$ lies on the line segment $\overline{AB}$ such that $AC:CB = 1:3$. What are the coordinates of point $C$?
   Answer: $(4, 0)$ Point $C$ divides $\overline{AB}$ in the ratio $1:3$, meaning $C$ is $\frac{1}{1+3} = \frac{1}{4}$ of the way from $A$ to $B$. Using the section formula: $C = A + \frac{1}{4}(B - A) = (3, -2) + \frac{1}{4}((7, 6) - (3, -2)) = (3, -2) + \frac{1}{4}(4, 8) = (3, -2) + (1, 2) = (4, 0)$. Choice B gives the point $\frac{1}{8}$ of the way from $A$ to $B$. Choice C gives the midpoint. Choice D gives the point $\frac{3}{8}$ of the way from $A$ to $B$.
3. Consider the ordered triple of points $(0,0),(2,1),(4,2)$. If the pattern continues in the same way, what will be the y-coordinate of the point whose x-coordinate is 10?
   Answer: 5 When you see a sequence of coordinate points, look for patterns in how the x and y values change from one point to the next. This tests your ability to recognize linear relationships and extend patterns. Let's examine the given points: $(0,0), (2,1), (4,2)$. Notice that as the x-coordinate increases by 2 each time (0 → 2 → 4), the y-coordinate increases by 1 each time (0 → 1 → 2). This creates a constant ratio: for every 2 units the x-coordinate increases, the y-coordinate increases by 1. You can also think of this as a linear relationship where $y = \frac{1}{2}x$. When $x = 10$, we get $y = \frac{1}{2}(10) = 5$. Alternatively, continue the pattern: $(0,0), (2,1), (4,2), (6,3), (8,4), (10,5)$. The point with x-coordinate 10 has y-coordinate 5. Looking at the wrong answers: Choice A (3) would be correct if x = 6, suggesting you might have miscounted the pattern steps. Choice B (4) corresponds to x = 8, indicating you stopped one step too early. Choice D (6) might result from incorrectly thinking the y-coordinate equals half the x-coordinate plus 1, or from some other calculation error. Strategy tip: For coordinate pattern problems, always identify the relationship between consecutive points first. Check if x and y values change by constant amounts (linear) or if there's a consistent ratio. When in doubt, extend the pattern step-by-step rather than jumping to a formula.
4. The coordinate plane shown displays a pattern where each point $(x_n, y_n)$ satisfies $x_{n+1} = x_n + 2$ and $y_{n+1} = y_n + x_n$. If $(x_1, y_1) = (1, 0)$, what are the coordinates of the 6th point?
   Answer: $(11, 25)$ Compute: $(1,0) \to (3, 0+1=1) \to (5, 1+3=4) \to (7, 4+5=9) \to (9, 9+7=16) \to (11, 16+9=25)$. So $P_6=(11,25)$. Choice B adds $y_n + x_{n+1}$. Choice C uses 7 points. Choice D computes cumulative sum of $x$'s incorrectly.
5. Based on the pattern shown in the coordinate plane, point $P_n$ has coordinates $(n, f(n))$. The first six points are $(1,2), (2,5), (3,10), (4,17), (5,26), (6,37)$. What is the $y$-coordinate of $P_{12}$ MINUS the $y$-coordinate of $P_7$?
   Answer: $95$ Pattern: $y=n^2+1$. $P_{12}$: $144+1=145$. $P_7$: $49+1=50$. Difference: $145-50=95$. Choice B uses $n^2$ without adding 1: $144-49=95$ (actually gives same answer, so this represents a different error). Choice C uses $P_{13}-P_7$: $170-50=120$ (not listed, so represents calculation error). Choice D rounds to nearest hundred.
6. In the coordinate plane shown, points $Q_1, Q_2, Q_3, \ldots$ are plotted at $(2,1), (4,4), (6,9), (8,16), \ldots$ If this pattern continues, at what $x$-coordinate does the $y$-value first exceed $200$?
   Answer: $30$ The pattern: $x=2n$, $y=n^2$, so $y=(x/2)^2$. Need $(x/2)^2 > 200$, so $x/2 > \sqrt{200} \approx 14.14$, $x/2 \geq 15$, $x \geq 30$. Check: $x=30$, $y=225 > 200$ ✓. $x=28$, $y=196 < 200$. Choice B gives $196$, just under. Choices C and D are off.
7. The points $(1,3),(3,5),(5,7),(7,9)$ follow a consistent pattern. Which point comes next in this pattern?
   Answer: $(9,11)$ When you encounter ordered pairs that "follow a consistent pattern," you're looking for a relationship between the x-coordinates and y-coordinates, and how both change from point to point. Let's examine the pattern systematically. Looking at the x-coordinates: $1, 3, 5, 7$. Each increases by 2, so the next x-coordinate should be $7 + 2 = 9$. Now for the y-coordinates: $3, 5, 7, 9$. These also increase by 2 each time, so the next y-coordinate should be $9 + 2 = 11$. You can also notice that in each point, the y-coordinate equals the x-coordinate plus 2: $(1,3)$ gives us $1 + 2 = 3$, $(3,5)$ gives us $3 + 2 = 5$, and so on. This confirms our pattern. Therefore, the next point is $(9,11)$, which is choice A. Let's check why the other options are incorrect. Choice B $(9,13)$ has the correct x-coordinate but jumps the y-coordinate by 4 instead of 2, breaking the consistent pattern. Choice C $(8,11)$ has the correct y-coordinate but uses 8 for x, which would mean adding 1 instead of 2 to the previous x-coordinate. Choice D $(11,9)$ reverses the relationship entirely and doesn't follow either sequence correctly. Strategy tip: For pattern questions, always check both coordinates separately and look for relationships between them. The SHSAT often includes distractors that follow one part of the pattern correctly but miss the other part.
8. The coordinate plane shown displays the first four points of a pattern: $(1, 2)$, $(2, 5)$, $(3, 10)$, and $(4, 17)$. If the pattern continues, what will be the $y$-coordinate of the point whose $x$-coordinate is $9$?
   Answer: $82$ The $y$-values follow the rule $y = x^2 + 1$: $1+1=2$, $4+1=5$, $9+1=10$, $16+1=17$. For $x=9$: $81+1=82$. Choice A ($65 = 8^2+1$) uses $x=8$. Choice C ($80 = 9^2-1$) uses the wrong sign. Choice D ($26 = 5^2+1$) extends only one more term.
9. The coordinate plane below shows a sequence of points: $(0,1), (1,2), (2,4), (3,8), (4,16)$. If the pattern continues, which point lies on the $x$-axis line $y = 1024$?
   Answer: $(10, 1024)$ Pattern: $y=2^x$. Solving $2^x=1024$: $x=10$. Choice B uses $y=2^{x+1}$. Choice C uses $y=2^{x-1}$. Choice D gives $2^8=256$.
10. Refer to the coordinate plane. Five points form a pattern such that each successive point is obtained by moving $+2$ in $x$ and doubling the previous $y$-value. If $P_1 = (1, 3)$, what is the sum of the coordinates of $P_5$?
    Answer: $57$ $P_1=(1,3)$, $P_2=(3,6)$, $P_3=(5,12)$, $P_4=(7,24)$, $P_5=(9,48)$. Sum = $9+48=57$. Choice B is the sum for $P_4$: $7+24=31$ (closest distractor). Choice C doubles both coordinates. Choice D uses $P_3$: $5+12=17$ (closest distractor).
11. The coordinate plane shown displays points $A_1, A_2, A_3, \ldots$ located at $(1,1), (2,4), (3,9), (4,16), \ldots$ Which of the following is NOT a point in this pattern?
    Answer: $(12, 140)$ The pattern is $y=x^2$. Check each: $7^2=49$ ✓, $10^2=100$ ✓, $12^2=144 \ne 140$ ✗, $15^2=225$ ✓.
12. Refer to the coordinate plane shown. The points follow a pattern in the order labeled $P_1, P_2, P_3, P_4, \ldots$. What are the coordinates of $P_{20}$?
    Answer: $(20, 39)$ The points are $P_1=(1,1)$, $P_2=(2,3)$, $P_3=(3,5)$, $P_4=(4,7)$. The pattern is $P_n=(n, 2n-1)$. For $n=20$: $(20, 39)$. Choice A uses $y=2n$. Choice B uses $x=n-1$. Choice D uses $n=21$.
13. Refer to the figure. The points plotted form a pattern on the coordinate plane: $(1,1), (2,3), (4,9), (8,27), (16,81), \ldots$ If the pattern continues, what is the $y$-coordinate of the point whose $x$-coordinate is $128$?
    Answer: $2187$ $x$-values: $1, 2, 4, 8, 16, \ldots = 2^{n-1}$. $y$-values: $1, 3, 9, 27, 81, \ldots = 3^{n-1}$. For $x=128=2^7$, $n-1=7$, so $y=3^7=2187$. Choice B is $3^6=729$ (off by one). Choice C is $3^8=6561$. Choice D is $3^5=243$.
14. The table lists coordinate points generated by a rule. Use the table to answer the question. Which point also satisfies the same rule?
    Answer: $(6, 19)$ From the table, the points are $(1,-1), (2,3), (3,7), (4,11), (5,15)$. The $y$-values increase by $4$: rule is $y = 4x - 5$. For $x=6$: $y=19$. Choice B adds 6 instead of 4. Choice C uses $y=3x$. Choice D uses $y=4x$.
15. A set of points follows the pattern: $(1, 1)$, $(2, 3)$, $(3, 6)$, $(4, 10)$, $(5, 15)$, $\ldots$ If point $(n, y)$ belongs to this set, which equation best describes the relationship between $n$ and $y$?
    Answer: $y = \frac{n(n+1)}{2}$ Examine the y-values: 1, 3, 6, 10, 15. These are triangular numbers. The differences between consecutive terms are 2, 3, 4, 5, which increase by 1 each time. This confirms the triangular number pattern $y = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$. Verify: For $n=1$: $y = \frac{1(2)}{2} = 1$ ✓. For $n=2$: $y = \frac{2(3)}{2} = 3$ ✓. For $n=3$: $y = \frac{3(4)}{2} = 6$ ✓. Choice B gives: for $n=1$, $y = 1 - 1 + 1 = 1$ ✓, but for $n=2$, $y = 4 - 2 + 1 = 3$ ✓, and for $n=3$, $y = 9 - 3 + 1 = 7 \neq 6$. Choice C: for $n=1$, $y = 2 - 1 = 1$ ✓, but for $n=2$, $y = 4 - 2 = 2 \neq 3$. Choice D: for $n=1$, $y = \frac{1(3)}{3} = 1$ ✓, but for $n=2$, $y = \frac{2(4)}{3} = \frac{8}{3} \neq 3$.

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