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{ "quality": "blurry", "question_count": 10, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Coordinate Geometry", "Linear Relationships", "Vertical Lines" ], "new_concepts": [ "Linear Intercepts", "Graphing Linear Equations" ], "current_concepts": [ "Coordinate Geometry", "Linear Relationships", "Vertical Lines", "Linear Intercepts", "Graphing Linear Equations" ] } ## Part A Question 1 Using the *Coordinate Geometry* knowledge point $$ \begin{aligned} &P = (3, -4)\ &y\text{-coordinate} = -4 \end{aligned} $$ ## Part A Question 2 Using the *Linear Relationships* knowledge point $$ \begin{aligned} &y = 3x^2 + 4 \quad (\text{quadratic})\ &y = 3^x \quad (\text{exponential})\ &y = \frac{3}{x} \quad (\text{hyperbolic})\ &y = 3x \quad (\text{linear}) \end{aligned} $$ ## Part A Question 3 Using the *Vertical Lines* knowledge point $$ \begin{aligned} &x = c\ &x = 5 \end{aligned} $$ ## Part A Question 4 Using the *Linear Relationships* knowledge point $$ \begin{aligned} &\text{rise} = 6, \quad \text{run} = 2\ &m = \frac{\text{rise}}{\text{run}} = \frac{6}{2} = 3 \end{aligned} $$ ## Part A Question 5 Using the *Linear Relationships* knowledge point $$ \begin{aligned} &y = 2x - 7\ &b = -7 \end{aligned} $$ ## Part B Question 1a Using the *Linear Relationships* knowledge point $$ \begin{aligned} &y = 3x\ &x = -1 \implies y = -3\ &x = 0 \implies y = 0\ &x = 1 \implies y = 3\ &x = 2 \implies y = 6 \end{aligned} $$ ## Part B Question 1b Using the *Linear Relationships* knowledge point $$ \begin{aligned} &y = -3x - 1\ &x = -1 \implies y = 2\ &x = 0 \implies y = -1\ &x = 1 \implies y = -4\ &x = 2 \implies y = -7 \end{aligned} $$ ## Part B Question 2a Using the *Coordinate Geometry* and *Linear Relationships* knowledge points $$ \begin{aligned} &(x, y) = (2, 8)\ &y = 2x + 6 \implies 8 = 2(2) + 6 = 10 \quad (\text{False}) \end{aligned} $$ ## Part B Question 2b Using the *Coordinate Geometry* and *Linear Relationships* knowledge points $$ \begin{aligned} &(x, y) = (2, 8)\ &y = 3x + 2 \implies 8 = 3(2) + 2 = 8 \quad (\text{True}) \end{aligned} $$ ## Part B Question 3 We find the intercepts for the linear relationship $ y = 2x + 8 $. - **y-intercept**: Substitute $ x = 0 $ into the equation: $$ y = 2(0) + 8 = 8 $$ - **x-intercept**: Substitute $ y = 0 $ into the equation: $$ 0 = 2x + 8 \implies 2x = -8 \implies x = -4 $$ ## Part B Question 4 We sketch the graph of the line $ y = 2x - 4 $. - **y-intercept**: Substitute $ x = 0 $: $$ y = 2(0) - 4 = -4 \implies (0, -4) $$ - **x-intercept**: Substitute $ y = 0 $: $$ 0 = 2x - 4 \implies 2x = 4 \implies x = 2 \implies (2, 0) $$ - We plot these two points and draw a straight line passing through them. ## Part A: Multiple-choice questions ### Question 1 What is the $y$-coordinate of the point $(3, -4)$? * (A) 3 * (B) 0 * (C) -4 * (D) -3 **Correct Answer:** (C) -4 --- ### Question 2 Which of the following equations represents a linear relationship? * (A) $y = 3x^2 + 4$ * (B) $y = 3^x$ * (C) $y = 3x$ * (D) $y = \frac{3}{x}$ **Correct Answer:** (C) $y = 3x$ --- ### Question 3 Which equation describes a vertical line with an $x$-intercept of 5? * (A) $y = 5$ * (B) $x + y = 5$ * (C) $y = 5x$ * (D) $x = 5$ **Correct Answer:** (D) $x = 5$ --- ### Question 4 If a line rises 6 units for every 2 units to the right, what is its gradient (slope)? * (A) 2 * (B) 6 * (C) 3 * (D) 12 **Correct Answer:** (C) 3 --- ### Question 5 For the line with the equation $y = 2x - 7$, what is the $y$-intercept ($b$)? * (A) -2 * (B) 0 * (C) -7 * (D) 7 **Correct Answer:** (C) -7 --- ## Part B: Short-answer questions ### Question 1 Complete the table of values for the linear rule: **a) $y = 3x$** | $x$ | -1 | 0 | 1 | 2 | |---|---|---|---|---| | $y$ | -3 | 0 | 3 | 6 | **b) $y = -3x - 1$** | $x$ | -1 | 0 | 1 | 2 | |---|---|---|---|---| | $y$ | 2 | -1 | -4 | -7 | --- ### Question 2 Determine if the point $(2, 8)$ lies on the following lines: **a) $y = 2x + 6$** No, because $8 eq 2(2) + 6$. **b) $y = 3x + 2$** Yes, because $8 = 3(2) + 2$. --- ### Question 3 For the linear relationship $y = 2x + 8$: **a) Find the $y$-intercept by substituting $x = 0$:** $y = 8$ **b) Find the $x$-intercept by substituting $y = 0$:** $x = -4$ --- ### Question 4 On a Cartesian plane: **a) Sketch the graph of the line $y = 2x - 4$** *(See the plotted graph below)* **b) Write the $x$-intercept:** $2$ **c) Write the $y$-intercept:** $-4$ { "elements": [ { "type": "functiongraph", "params": [ { "js": "2*x - 4", "latex": "2x - 4" }, -10, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "y = 2x - 4", "withLabel": true } }, { "type": "point", "params": [ [2, 0] ], "properties": { "name": "(2, 0)", "size": 4, "strokeColor": "#F2557F", "fillColor": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [0, -4] ], "properties": { "name": "(0, -4)", "size": 4, "strokeColor": "#F2557F", "fillColor": "#F2557F", "withLabel": true } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Coordinate Geometry" ] }

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Part A Question 1

Part A Question 2

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Part A Question 4

Part A Question 5

Part B Question 1a

Part B Question 1b

Part B Question 2a

Part B Question 2b

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Answer:

Part A: Multiple-choice questions

Question 1

Question 2

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Question 4

Question 5

Part B: Short-answer questions

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Question 4