Question

Graph function then state the domain and range of f(x)=x^2+6x+8

Answer 1

Answer:



Explanation:

To graph the function f(x) = x^2 + 6x + 8, we can start by finding the vertex and the axis of symmetry. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.

In this case, we have f(x) = x^2 + 6x + 8. To convert this into vertex form, we can complete the square.

First, let's group the terms with x:
f(x) = (x^2 + 6x) + 8.

Next, we add and subtract the square of half the coefficient of
x (6/2 = 3)^2 = 9
f(x) = (x^2 + 6x + 9 - 9) + 8.

Now, we can rewrite the expression as a perfect square: f(x) = (x^2 + 6x + 9) - 9 + 8.

Simplifying further, we have
f(x) = (x + 3)^2 - 1.

From this equation, we can see that the vertex of the parabola is at (-3, -1). The axis of symmetry is the vertical line that passes through the vertex, which in this case is x = -3.

To determine the domain and range of the function, we consider the x-values and y-values that the function can take.

The domain represents all possible x-values for which the function is defined. In this case, since the function is quadratic, it is defined for all real numbers. Therefore, the domain is (-infinitely, infinitely).

The range represents all possible y-values that the function can take. By analyzing the graph, we can see that the vertex is the lowest point of the parabola, and it opens upwards. This means that the range is all real numbers greater than or equal to the y-coordinate of the vertex. In this case, the range is [-1, infinitely).

Answer 2

• graph: see attached
• domain: all real numbers
• range: y ≥ -1

Explanation:

You want the graph and domain and range of the function f(x) = x² +6x +8.

Vertex

We can rewrite the function in vertex form as ...

f(x) = (x² +6x +9) -1

f(x) = (x +3)² -1

Comparing this to the vertex form equation f(x) = (x -h)² +k, we see that (h, k) = (-3, -1). This is the vertex of the parabola, marked on the attached graph.

Domain

The domain of f(x) is the horizontal extent of the graph. The function is defined for all possible values of x, so ...

The domain is all real numbers.

Range

The range of f(x) is the vertical extent of the graph. The function has a minimum value of -1, so ...

The range is all values of y greater than or equal to -1.

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