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 $y \leq x^2 - 4x + 1$
y \geq x^2 + 4
y = -x^2 + 4
solid curve
1st point: $(0,0)$
y = -x^2 + 4
$0 = -(0)^2 + 4$
$0 \neq 4$ false
Solve for the x-intercept
$0 = x^2 + 4$
$x^2 = \pm \sqrt{4}$
$x = \pm 2$
$x_1 = 2$
$(2,0)$
$x_2 = -2$
$(-2,0)$
Find the vertex
$V(h,k) = \left(-\frac{b}{2a}, \frac{4ac-b^2}{4a}\right)$
a = -1
b = 0
c = 4
= $(0,4)$
Graph shape: a downward-opening parabola with vertex at $(0,4)$, x-intercepts at $(-2,0)$ and $(2,0)$, and shaded regions outside the parabola. position_hint: center
Solve the following using the solution provided
1. **State the problem:** We are asked to analyze and graph the parabola given by the inequality $y \geq x^2 + 4$ and the equation $y = -x^2 + 4$. We will find the vertex, x-intercepts, and describe the shading region.
2. **Find the x-intercepts of $y = -x^2 + 4$:**
Set $y=0$ to find x-intercepts:
$$0 = -x^2 + 4$$
Rearranged:
$$x^2 = 4$$
Taking square roots:
$$x = \pm 2$$
So the x-intercepts are at points $(2,0)$ and $(-2,0)$.
3. **Find the vertex of the parabola $y = -x^2 + 4$:**
The vertex formula for a parabola $y = ax^2 + bx + c$ is:
$$V\left(h,k\right) = \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$$
Here, $a = -1$, $b = 0$, and $c = 4$.
Calculate $h$:
$$h = -\frac{0}{2 \times -1} = 0$$
Calculate $k$:
$$k = 4 - \frac{0^2}{4 \times -1} = 4$$
So the vertex is at $(0,4)$.
4. **Graph shape and shading:**
Since $a = -1 < 0$, the parabola opens downward.
The inequality $y \geq x^2 + 4$ describes the region above the parabola $y = x^2 + 4$, which opens upward with vertex at $(0,4)$.
The parabola $y = -x^2 + 4$ opens downward with vertex at $(0,4)$ and x-intercepts at $(-2,0)$ and $(2,0)$.
The shaded region for $y = -x^2 + 4$ is outside the parabola (above it), as described.
5. **Summary:**
- Parabola $y = -x^2 + 4$ has vertex $(0,4)$ and x-intercepts $(-2,0)$ and $(2,0)$.
- It opens downward.
- The region outside the parabola is shaded.
This completes the analysis and graphing of the given parabola and inequalities.