Final Answer:
Graph 1 → y = (x + 2)(x – 4)
Graph 2 → y = (x – 2)² – 4
Graph 3 → y = –(x + 2)² + 4
Graph 4 → y = ½x² + x – 4
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Explanation:
Let’s go step by step to match each graph with the right equation.
🔹 Graph 1 (first from left):
- It opens upward.
- It crosses the x-axis at x = –2 and x = 4.
- That means the roots are –2 and 4.
- The equation y = (x + 2)(x – 4) gives exactly those roots.
✔ So, this graph matches y = (x + 2)(x – 4).
🔹 Graph 2 (second from left):
- It opens upward.
- Its vertex is at (2, –4).
- The equation y = (x – 2)² – 4 has vertex (2, –4) — that’s perfect!
✔ So, this graph matches y = (x – 2)² – 4.
🔹 Graph 3 (third from left):
- It opens downward (sad face).
- Its vertex is at (–2, 4).
- The equation y = –(x + 2)² + 4 has a negative sign (so it opens down) and vertex (–2, 4).
✔ So, this graph matches y = –(x + 2)² + 4.
🔹 Graph 4 (last one):
- It opens upward.
- It crosses the x-axis near x = –4 and x = 2.
- Let’s test y = ½x² + x – 4.
If x = 0, y = –4 → y-intercept is (0, –4).
If x = 2: y = ½(4) + 2 – 4 = 2 + 2 – 4 = 0 → so (2, 0) is on the graph.
If x = –4: y = ½(16) – 4 – 4 = 8 – 4 – 4 = 0 → so (–4, 0) is on the graph.
✔ This matches the last graph.
So we’ve matched all four!
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Parent Tip: Review the logic above to help your child master the concept of characteristics of quadratic functions worksheet.
