# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-5/12

Exercises: Try Out - Set 5: Real exam simulation to sharpen your skills and build confidence. - Problem 12

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## Exercise 12

### Question

import { QuestionGraph } from "../graph";

export const metadata = {
  title: "Problem 12",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Consider the graph below!

<QuestionGraph
  title="Parabola Graph"
  description={
    <>
      A parabola opening to the left intersecting the <InlineMath math="y" />-axis at <InlineMath math="4" /> and <InlineMath math="-2" />, and the <InlineMath math="x" />-axis at <InlineMath math="8" />.
    </>
  }
/>

The correct statements are:

1. One of the intersection points with the <InlineMath math="y" />-axis is <InlineMath math="(0, -2)" />.
2. <InlineMath math="f(y) = -(y-1)^2 + 9" />.
3. Axis of symmetry <InlineMath math="y = 1" />.
4. Intersection point with the <InlineMath math="x" />-axis is <InlineMath math="(8, 0)" />.


### Choices

- [ ] If $$(1)$$, $$(2)$$, and $$(3)$$ are correct.
- [ ] If $$(1)$$ and $$(3)$$ are correct.
- [ ] If $$(2)$$ and $$(4)$$ are correct.
- [ ] If only $$(4)$$ is correct.
- [x] If all are correct.

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 12",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We analyze each statement based on the graph:

1. **Statement <InlineMath math="(1)" />**: The graph intersects the <InlineMath math="y" />-axis at <InlineMath math="4" /> and <InlineMath math="-2" />. So the intersection points are <InlineMath math="(0, 4)" /> and <InlineMath math="(0, -2)" />. Statement <InlineMath math="(1)" /> is **CORRECT**.

2. **Statement <InlineMath math="(2)" />**:
   Since the graph intersects the <InlineMath math="y" />-axis at <InlineMath math="y = 4" /> and <InlineMath math="y = -2" />, the equation is of the form:
   <MathContainer>
     <BlockMath math="x = a(y - 4)(y + 2)" />
   </MathContainer>
   The graph passes through the point <InlineMath math="(8, 0)" /> (x-intercept). Substitute into the equation:
   <MathContainer>
     <BlockMath math="8 = a(0 - 4)(0 + 2)" />
     <BlockMath math="8 = a(-4)(2)" />
     <BlockMath math="8 = -8a \implies a = -1" />
   </MathContainer>
   So the function equation is:
   <MathContainer>
     <BlockMath math="x = -(y - 4)(y + 2)" />
     <BlockMath math="x = -(y^2 - 2y - 8)" />
     <BlockMath math="x = -y^2 + 2y + 8" />
   </MathContainer>
   Converting to vertex form:
   <MathContainer>
     <BlockMath math="x = -(y^2 - 2y + 1 - 1 - 8)" />
     <BlockMath math="x = -((y - 1)^2 - 9)" />
     <BlockMath math="x = -(y - 1)^2 + 9" />
   </MathContainer>
   Or in function notation <InlineMath math="f(y)" />:
   <MathContainer>
     <BlockMath math="f(y) = -(y - 1)^2 + 9" />
   </MathContainer>
   Statement <InlineMath math="(2)" /> is **CORRECT**.

3. **Statement <InlineMath math="(3)" />**:
   The axis of symmetry of a horizontal parabola is <InlineMath math="y_p" /> (ordinate of the vertex).
   From the form <InlineMath math="x = a(y - y_p)^2 + x_p" />, we get <InlineMath math="y_p = 1" />.
   Or from the midpoint of the roots:
   <MathContainer>
     <BlockMath math="y = \frac{4 + (-2)}{2} = \frac{2}{2} = 1" />
   </MathContainer>
   So the axis of symmetry is <InlineMath math="y = 1" />. Statement <InlineMath math="(3)" /> is **CORRECT**.

4. **Statement <InlineMath math="(4)" />**:
   The intersection with the <InlineMath math="x" />-axis occurs when <InlineMath math="y = 0" />.
   From the graph, it clearly intersects at <InlineMath math="x = 8" />.
   By calculation: <InlineMath math="x = -(0 - 1)^2 + 9 = -1 + 9 = 8" />.
   So the intersection point is <InlineMath math="(8, 0)" />. Statement <InlineMath math="(4)" /> is **CORRECT**.

Conclusion: All statements <InlineMath math="(1)" />, <InlineMath math="(2)" />, <InlineMath math="(3)" />, and <InlineMath math="(4)" /> are correct.


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