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{ "quality": "clear", "question_count": 10, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Quadratic Vertex Form", "Maximum Minimum Value", "Determining Quadratic Equations" ], "new_concepts": [ "Number of x-intercepts", "Standard to Vertex Form Conversion" ], "current_concepts": [ "Quadratic Vertex Form", "Maximum Minimum Value", "Determining Quadratic Equations", "Number of x-intercepts", "Standard to Vertex Form Conversion" ] } The image is upside down. Let's orient ourselves and solve the visible questions. The questions are numbered 6 through 14. Let's list each question clearly: ### Section 1: Determine the vertex form equation for the following parabolas. * **Question 6**: A quadratic function has a vertex at $(9, -3)$ and passes through $(4, 97)$. * **Question 7**: A quadratic function has a vertex at $(-7, 4)$ and passes through the point $(3, 54)$. * **Question 8**: A quadratic function has a vertex at $(3, -12)$ and passes through the point $(15, 96)$. * **Question 9**: A quadratic function has a vertex at $(10, 17)$ and a y-intercept of $16$. * **Question 10**: A quadratic function has a maximum value of $98$ and x-intercepts at $(-6, 0)$ and $(6, 0)$. ### Section 2: Without Graphing, Determine the vertex, max or min value, and number of x ints. * **Question 11**: $y = -7(x + 41)^2 - 2$ * **Question 12**: $y = 14.2(x - 17)^2$ * **Question 13**: $y = x^2 - 8x + 17$ * **Question 14**: $y = -\frac{1}{6}x^2 + 3x - \frac{40}{3}$ --- ## Find vertex form equations (Questions 6 to 10) Using the **Quadratic Vertex Form** knowledge point: The vertex form is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Using the **Determining Quadratic Equations** knowledge point: We substitute the vertex $(h, k)$ and the given point $(x, y)$ to solve for $a$. ### Question 6 Using the *Quadratic Vertex Form* and *Determining Quadratic Equations* knowledge points Vertex $(h, k) = (9, -3)$, point $(x, y) = (4, 97)$. $$ \begin{aligned} y &= a(x - 9)^2 - 3 \ 97 &= a(4 - 9)^2 - 3 \ 100 &= 25a \ a &= 4 \end{aligned} $$ Equation: $y = 4(x - 9)^2 - 3$ ### Question 7 Using the *Quadratic Vertex Form* and *Determining Quadratic Equations* knowledge points Vertex $(h, k) = (-7, 4)$, point $(x, y) = (3, 54)$. $$ \begin{aligned} y &= a(x + 7)^2 + 4 \ 54 &= a(3 + 7)^2 + 4 \ 50 &= 100a \ a &= 0.5 \end{aligned} $$ Equation: $y = 0.5(x + 7)^2 + 4$ ### Question 8 Using the *Quadratic Vertex Form* and *Determining Quadratic Equations* knowledge points Vertex $(h, k) = (3, -12)$, point $(x, y) = (15, 96)$. $$ \begin{aligned} y &= a(x - 3)^2 - 12 \ 96 &= a(15 - 3)^2 - 12 \ 108 &= 144a \ a &= \frac{108}{144} = 0.75 \end{aligned} $$ Equation: $y = 0.75(x - 3)^2 - 12$ ### Question 9 Using the *Quadratic Vertex Form* and *Determining Quadratic Equations* knowledge points Vertex $(h, k) = (10, 17)$, y-intercept is $16$, which means it passes through $(0, 16)$. $$ \begin{aligned} y &= a(x - 10)^2 + 17 \ 16 &= a(0 - 10)^2 + 17 \ -1 &= 100a \ a &= -0.01 \end{aligned} $$ Equation: $y = -0.01(x - 10)^2 + 17$ ### Question 10 Using the *Quadratic Vertex Form* and *Determining Quadratic Equations* knowledge points The x-intercepts are symmetric at $-6$ and $6$, so the axis of symmetry is $x = 0$. The vertex is at $(0, 98)$ since the maximum value is $98$. Using the point $(6, 0)$: $$ \begin{aligned} y &= a(x - 0)^2 + 98 \ 0 &= a(6)^2 + 98 \ -98 &= 36a \ a &= -\frac{98}{36} = -\frac{49}{18} \end{aligned} $$ Equation: $y = -\frac{49}{18}x^2 + 98$ --- ## Analyze properties without graphing (Questions 11 to 14) Using the **Maximum Minimum Value** knowledge point: For $y = a(x - h)^2 + k$: - The vertex is $(h, k)$. - If $a > 0$, the minimum value is $k$. - If $a < 0$, the maximum value is $k$. - The number of x-intercepts depends on the signs of $a$ and $k$: - If $k = 0$, there is exactly $1$ x-intercept (at the vertex). - If $a$ and $k$ have opposite signs, there are $2$ x-intercepts. - If $a$ and $k$ have the same sign, there are $0$ x-intercepts. ### Question 11 Using the *Maximum Minimum Value* knowledge point Equation: $y = -7(x + 41)^2 - 2$ - Vertex: $(-41, -2)$ - Since $a = -7 < 0$, it has a maximum value of $-2$. - Since $a < 0$ and $k < 0$ (same sign), there are $0$ x-intercepts. ### Question 12 Using the *Maximum Minimum Value* knowledge point Equation: $y = 14.2(x - 17)^2$ - Vertex: $(17, 0)$ - Since $a = 14.2 > 0$, it has a minimum value of $0$. - Since $k = 0$, there is $1$ x-intercept. ### Question 13 ## Convert standard to vertex form We convert $y = x^2 - 8x + 17$ by completing the square: $$ \begin{aligned} y &= (x^2 - 8x + 16) - 16 + 17 \ y &= (x - 4)^2 + 1 \end{aligned} $$ ## Determine properties for Question 13 Using the *Maximum Minimum Value* knowledge point - Vertex: $(4, 1)$ - Since $a = 1 > 0$, it has a minimum value of $1$. - Since $a > 0$ and $k > 0$ (same sign), there are $0$ x-intercepts. ### Question 14 ## Convert standard to vertex form for Question 14 We convert $y = -\frac{1}{6}x^2 + 3x - \frac{40}{3}$: $$ \begin{aligned} y &= -\frac{1}{6}(x^2 - 18x) - \frac{40}{3} \ y &= -\frac{1}{6}(x^2 - 18x + 81) + \frac{81}{6} - \frac{40}{3} \ y &= -\frac{1}{6}(x - 9)^2 + \frac{27}{2} - \frac{80}{6} \ y &= -\frac{1}{6}(x - 9)^2 + \frac{81 - 80}{6} \ y &= -\frac{1}{6}(x - 9)^2 + \frac{1}{6} \end{aligned} $$ ## Determine properties for Question 14 Using the *Maximum Minimum Value* knowledge point - Vertex: $(9, \frac{1}{6})$ - Since $a = -\frac{1}{6} < 0$, it has a maximum value of $\frac{1}{6}$. - Since $a < 0$ and $k > 0$ (opposite signs), there are $2$ x-intercepts. | No. | Answer | |---|---| | 6 | $y = 4(x - 9)^2 - 3$ | | 7 | $y = 0.5(x + 7)^2 + 4$ | | 8 | $y = 0.75(x - 3)^2 - 12$ | | 9 | $y = -0.01(x - 10)^2 + 17$ | | 10 | $y = -\frac{49}{18}x^2 + 98$ | | 11 | Vertex: $(-41, -2)$, Max value: $-2$, Number of x-intercepts: $0$ | | 12 | Vertex: $(17, 0)$, Min value: $0$, Number of x-intercepts: $1$ | | 13 | Vertex: $(4, 1)$, Min value: $1$, Number of x-intercepts: $0$ | | 14 | Vertex: $(9, \frac{1}{6})$, Max value: $\frac{1}{6}$, Number of x-intercepts: $2$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Vertex Form" ] }

No.	Answer
7	\(y = 0.5(x + 7)^2 + 4\)
8	\(y = 0.75(x - 3)^2 - 12\)
9	\(y = -0.01(x - 10)^2 + 17\)
10	\(y = -\frac{49}{18}x^2 + 98\)
11	Vertex: \((-41, -2)\), Max value: \(-2\), Number of x-intercepts: \(0\)
12	Vertex: \((17, 0)\), Min value: \(0\), Number of x-intercepts: \(1\)
13	Vertex: \((4, 1)\), Min value: \(1\), Number of x-intercepts: \(0\)
14	Vertex: \((9, \frac{1}{6})\), Max value: \(\frac{1}{6}\), Number of x-intercepts: \(2\)

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Explanation:

Section 1: Determine the vertex form equation for the following parabolas.

Section 2: Without Graphing, Determine the vertex, max or min value, and number of x ints.

Find vertex form equations (Questions 6 to 10)

Question 6

Question 7

Question 8

Question 9

Question 10

Answer: