) Sketch the graphs of the line y = 2 + x and the parabola y = 2x
2 − 3x − 31 on
the same set of axis. Show all working including finding the x- and y-intercepts,
and the co-ordinates where the two graphs intersect. Shade the area enclosed by
the parabola and the line.

Here is the graph showing both the line \( y = 2 + x \) (blue) and the parabola \( y = 2x^2 - 3x - 31 \) (red). The shaded region represents the area enclosed between the parabola and the line.

### Key Points:
- The line intersects the x-axis at \( (-2, 0) \) and the y-axis at \( (0, 2) \).
- The parabola has its y-intercept at \( (0, -31) \), and its x-intercepts are approximately \( \frac{3 \pm \sqrt{257}}{4} \).
- The intersection points of the line and parabola are approximately \( (6.69, 8.69) \) and \( (-4.69, -2.69) \).
- The gray shaded region represents the enclosed area between the two curves.

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### Related Questions:
1. How do you calculate the area enclosed by two curves?
2. What are the properties of a parabola in general form \( y = ax^2 + bx + c \)?
3. How do you solve a quadratic equation using the discriminant?
4. What is the geometric significance of the intersection points between a line and a parabola?
5. How do you determine the direction (upward or downward) of a parabola?

### Tip:
When solving quadratic equations, the discriminant \( \Delta = b^2 - 4ac \) helps determine the nature of the solutions (real or complex, and distinct or repeated).

Sketch the graphs of the line y = 2 + x and the parabola y = 2x^2 − 3x − 31 on the same set of axes. Show all working including finding the x- and y-intercepts, and the co-ordinates where the two graphs intersect. Shade the area enclosed by the parabola and the line.

This detailed guide covers graphing the line y = 2 + x and the parabola y = 2x^2 - 3x - 31, including finding the x- and y-intercepts, calculating the intersection points, and shading the enclosed area. Suitable for students in Grades 10-12.

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