Characteristics of Quadratic Functions Worksheets - Math Monks - Free Printable

Let's analyze each quadratic graph and determine the required properties: Domain, Range, x-intercepts, y-intercept, Vertex, Maximum/Minimum value, Axis of Symmetry, and whether the parabola opens up or down.

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🔹 Graph 1

Observations:
- Parabola opens downward (since it has a maximum point).
- Vertex is at $(-2, 1)$.
- It crosses the x-axis at $x = -4$ and $x = 0$, so x-intercepts are $(-4, 0)$ and $(0, 0)$.
- y-intercept is at $(0, 0)$.
- The highest point is the vertex: $y = 1$, so maximum value is 1.
- Axis of symmetry is the vertical line through the vertex: $x = -2$.
- Domain: All real numbers (parabolas extend infinitely left and right).
- Range: From negative infinity up to the vertex: $(-\infty, 1]$

#### ✔ Answers for Graph 1:
- Domain: $(-\infty, \infty)$
- Range: $(-\infty, 1]$
- x-intercepts: $(-4, 0)$ and $(0, 0)$
- y-intercept: $(0, 0)$
- Vertex: $(-2, 1)$
- Maximum value: $1$
- Axis of symmetry: $x = -2$
- Open up or down: Down

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🔹 Graph 2

Observations:
- Opens downward (has a maximum).
- Vertex is at $(0, 2)$.
- Crosses x-axis at $x = -3$ and $x = 3$, so x-intercepts: $(-3, 0)$ and $(3, 0)$.
- y-intercept: $(0, 2)$ — same as vertex.
- Maximum value: $2$
- Axis of symmetry: $x = 0$ (the y-axis)
- Domain: $(-\infty, \infty)$
- Range: $(-\infty, 2]$

#### ✔ Answers for Graph 2:
- Domain: $(-\infty, \infty)$
- Range: $(-\infty, 2]$
- x-intercepts: $(-3, 0)$ and $(3, 0)$
- y-intercept: $(0, 2)$
- Vertex: $(0, 2)$
- Maximum value: $2$
- Axis of symmetry: $x = 0$
- Open up or down: Down

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🔹 Graph 3

Observations:
- Opens upward (has a minimum point).
- Vertex is at $(1, -2)$.
- Crosses x-axis at $x = -1$ and $x = 3$, so x-intercepts: $(-1, 0)$ and $(3, 0)$.
- y-intercept: when $x = 0$, $y = -1$, so $(0, -1)$.
- Minimum value: $-2$
- Axis of symmetry: $x = 1$
- Domain: $(-\infty, \infty)$
- Range: $[-2, \infty)$ (from minimum up)

#### ✔ Answers for Graph 3:
- Domain: $(-\infty, \infty)$
- Range: $[-2, \infty)$
- x-intercepts: $(-1, 0)$ and $(3, 0)$
- y-intercept: $(0, -1)$
- Vertex: $(1, -2)$
- Minimum value: $-2$ (Note: no maximum since it opens up)
- Axis of symmetry: $x = 1$
- Open up or down: Up

> ⚠️ Note: "Maximum value" is not applicable here — instead, we have a minimum value. But since the worksheet asks for "Maximum value", we write N/A or leave blank if not applicable.

But let’s follow the instruction: it says “Maximum value” — if there is no maximum, write none or N/A.

So:
- Maximum value: None

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🔹 Graph 4

Observations:
- Opens downward (has a maximum).
- Vertex is at $(-2, 4)$.
- Crosses x-axis at $x = -4$ and $x = 0$, so x-intercepts: $(-4, 0)$ and $(0, 0)$.
- y-intercept: $(0, 0)$
- Maximum value: $4$
- Axis of symmetry: $x = -2$
- Domain: $(-\infty, \infty)$
- Range: $(-\infty, 4]$

#### ✔ Answers for Graph 4:
- Domain: $(-\infty, \infty)$
- Range: $(-\infty, 4]$
- x-intercepts: $(-4, 0)$ and $(0, 0)$
- y-intercept: $(0, 0)$
- Vertex: $(-2, 4)$
- Maximum value: $4$
- Axis of symmetry: $x = -2$
- Open up or down: Down

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✔ Final Answer Summary:

#### Graph 1:
- Domain: $(-\infty, \infty)$
- Range: $(-\infty, 1]$
- x-intercepts: $(-4, 0), (0, 0)$
- y-intercept: $(0, 0)$
- Vertex: $(-2, 1)$
- Maximum value: $1$
- Axis of symmetry: $x = -2$
- Open up or down: Down

#### Graph 2:
- Domain: $(-\infty, \infty)$
- Range: $(-\infty, 2]$
- x-intercepts: $(-3, 0), (3, 0)$
- y-intercept: $(0, 2)$
- Vertex: $(0, 2)$
- Maximum value: $2$
- Axis of symmetry: $x = 0$
- Open up or down: Down

#### Graph 3:
- Domain: $(-\infty, \infty)$
- Range: $[-2, \infty)$
- x-intercepts: $(-1, 0), (3, 0)$
- y-intercept: $(0, -1)$
- Vertex: $(1, -2)$
- Maximum value: None
- Axis of symmetry: $x = 1$
- Open up or down: Up

#### Graph 4:
- Domain: $(-\infty, \infty)$
- Range: $(-\infty, 4]$
- x-intercepts: $(-4, 0), (0, 0)$
- y-intercept: $(0, 0)$
- Vertex: $(-2, 4)$
- Maximum value: $4$
- Axis of symmetry: $x = -2$
- Open up or down: Down

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✔ These answers are based on visual analysis of the graphs. Let me know if you'd like the equations derived from these graphs!