Question: Worked Examples
1 Graph
y \geq x^2 - 2
2 Graph
y < x^2 + 2x - 3
3 Graph
x^2 + y \leq 16
4 Graph
y > -x^2 + 6
5 Graph
y \leq x^2 - 4x + 1
top-left: a small sketch shows a parabola opening upward with axes labeled x and y, and the curve touching the x-axis near the origin.
top-right: a small sketch shows a parabola opening upward with axes labeled x and y, centered above the origin.
center: no separate graph shape shown; the main math expression is x^2 + y \leq 16.
bottom-left: no separate graph shape shown; the main math expression is y > -x^2 + 6.
bottom-right: no separate graph shape shown; the main math expression is y \leq x^2 - 4x + 1.
1. **Problem:** Graph the inequality $$y \geq x^2 - 2$$.
**Step 1:** Recognize this is a parabola opening upwards with vertex at $(0,-2)$.
**Step 2:** The inequality $$y \geq x^2 - 2$$ means the region on or above the parabola is shaded.
2. **Problem:** Graph the inequality $$y < x^2 + 2x - 3$$.
**Step 1:** Rewrite the quadratic in vertex form by completing the square:
$$x^2 + 2x - 3 = (x^2 + 2x + 1) - 1 - 3 = (x+1)^2 - 4$$
**Step 2:** The parabola opens upward with vertex at $(-1,-4)$.
**Step 3:** The inequality $$y < (x+1)^2 - 4$$ means the region strictly below the parabola is shaded.
3. **Problem:** Graph the inequality $$x^2 + y \leq 16$$.
**Step 1:** Solve for $$y$$:
$$y \leq 16 - x^2$$
**Step 2:** This is a downward opening parabola with vertex at $(0,16)$.
**Step 3:** The region on or below the parabola is shaded.
4. **Problem:** Graph the inequality $$y > -x^2 + 6$$.
**Step 1:** This is a downward opening parabola with vertex at $(0,6)$.
**Step 2:** The inequality $$y > -x^2 + 6$$ means the region strictly above the parabola is shaded.
5. **Problem:** Graph the inequality $$y \leq x^2 - 4x + 1$$.
**Step 1:** Complete the square to find vertex form:
$$x^2 - 4x + 1 = (x^2 - 4x + 4) - 4 + 1 = (x-2)^2 - 3$$
**Step 2:** The parabola opens upward with vertex at $(2,-3)$.
**Step 3:** The inequality $$y \leq (x-2)^2 - 3$$ means the region on or below the parabola is shaded.