Final Answer:
The function is piecewise:
- For $ x \leq -4 $, it's a line through $(-8, -6)$ and $(-4, -2)$: slope = $\frac{-2 - (-6)}{-4 - (-8)} = \frac{4}{4} = 1$, so $ y = x + 2 $.
- For $ x > -4 $, it's a parabola opening upward with vertex at $(0,0)$, passing through $(4,16)$: $ y = ax^2 $, plug in $(4,16)$ → $16 = a(16)$ → $a = 1$, so $ y = x^2 $.
But note the point at $x = -4$: the left piece gives $y = -4 + 2 = -2$ (included), and the right piece at $x = -4$ would be $(-4)^2 = 16$, but the graph shows a *hole* at $(-4, 4)$? Wait — re-examining coordinates:
From image (interpreting grid):
- Left red segment endpoints: $(-8, -6)$ and $(-4, -2)$ — both solid dots → included.
- Then a jump to point at $(-2, 4)$ — solid dot.
- Then curve goes down to $(0,0)$, up to $(4,16)$ — all solid.
Wait — there’s also a point at $(-2, 4)$, and the curve from $(-2,4)$ to $(0,0)$ to $(4,16)$ looks like part of a parabola. Let’s test if it's $y = x^2$:
- At $x = -2$, $x^2 = 4$ ✓
- At $x = 0$, $0^2 = 0$ ✓
- At $x = 4$, $4^2 = 16$ ✓
So the quadratic part is $y = x^2$, defined for $x \ge -2$? But what about between $-4 < x < -2$? The graph shows a gap — no points drawn there. The point at $(-2, 4)$ is included, and the left segment ends at $(-4, -2)$. So the domain is split:
- Segment: from $x = -8$ to $x = -4$, inclusive → line.
- Gap: $-4 < x < -2$
- Curve: from $x = -2$ onward → parabola $y = x^2$
But the problem likely asks:
What is the value of the function at $x = -2$?
Looking at the graph: at $x = -2$, there's a solid blue dot at $( -2, 4 )$. So $f(-2) = 4$.
Alternatively, maybe the question is: “Find $f(-2)$”.
Since no explicit question is given in the prompt, but the standard task with such a graph is often to evaluate $f(-2)$ or identify the piecewise definition.
Given typical homework style and the clear marked point at $(-2, 4)$, and that it's the only ambiguous point (junction), the most probable intended question is:
What is $f(-2)$?
Answer: 4.
Double-check:
- Left line: $y = x + 2$. At $x = -2$, that would give $y = 0$, but that point is *not* on the line segment (segment ends at $x = -4$).
- Parabola $y = x^2$: at $x = -2$, $y = 4$, and the dot is there → included.
Thus, $f(-2) = 4$.
Final Answer:
4
Parent Tip: Review the logic above to help your child master the concept of piecewise function worksheet.
