Graphing A Parabola From Vertex Form Worksheet - Fill and Sign ... - Free Printable

This image is a worksheet titled "Graphing a Parabolas from Vertex Form Worksheet." It presents four quadratic functions in vertex form, $y = a(x-h)^2 + k$, and asks the student to identify the vertex, the axis of symmetry (A.O.S.), and whether the vertex represents a maximum or minimum point.

I will solve each problem step-by-step by analyzing the given equation for each function. The vertex form of a parabola directly provides the vertex $(h, k)$ and the axis of symmetry, which is the vertical line $x = h$. The sign of the coefficient '$a$' determines if the parabola opens up (minimum) or down (maximum).

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Problem 1: $y = (x - 0)^2 + 2$

- Vertex: In the form $y = a(x-h)^2 + k$, we can see that $h = 0$ and $k = 2$. So, the vertex is $(0, 2)$.
- A.O.S.: The axis of symmetry is the line $x = h$, so it is $x = 0$.
- Max or Min? The coefficient $a = 1$, which is positive. This means the parabola opens upwards, so the vertex is a minimum.

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Problem 2: $y = 2(x - 3)^2 + 5$

- Vertex: Here, $h = 3$ and $k = 5$. So, the vertex is $(3, 5)$.
- A.O.S.: The axis of symmetry is $x = h$, so it is $x = 3$.
- Max or Min? The coefficient $a = 2$, which is positive. The parabola opens upwards, so the vertex is a minimum.

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Problem 3: $y = -3(x + 1)^2 - 4$

- Vertex: This equation can be rewritten as $y = -3(x - (-1))^2 - 4$. So, $h = -1$ and $k = -4$. The vertex is $(-1, -4)$.
- A.O.S.: The axis of symmetry is $x = h$, so it is $x = -1$.
- Max or Min? The coefficient $a = -3$, which is negative. The parabola opens downwards, so the vertex is a maximum.

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Problem 4: $y = (x - 5)^2 - 3$

- Vertex: Here, $h = 5$ and $k = -3$. So, the vertex is $(5, -3)$.
- A.O.S.: The axis of symmetry is $x = h$, so it is $x = 5$.
- Max or Min? The coefficient $a = 1$, which is positive. The parabola opens upwards, so the vertex is a minimum.

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Final Answer

Here are the completed answers for the worksheet:

1.
- Vertex: $(0, 2)$
- A.O.S.: $x = 0$
- Is the vertex a max or min? min

2.
- Vertex: $(3, 5)$
- A.O.S.: $x = 3$
- Is the vertex a max or min? min

3.
- Vertex: $(-1, -4)$
- A.O.S.: $x = -1$
- Is the vertex a max or min? max

4.
- Vertex: $(5, -3)$
- A.O.S.: $x = 5$
- Is the vertex a max or min? min