91Ó°ÊÓ

In mathematics, understanding the domain and range of a function is crucial for graphing and analyzing mathematical relationships. The **domain** of a function is the set of all possible input values (or 'x' values) for which the function is defined. For the quadratic function \( f(x) = (x+4)^2 \), there are no limitations on the values that \( x \) can take. This means the function can accept any real number as its input, making the domain

• \((-\infty, \infty)\)

On the other hand, the **range** of a function is the set of possible output values (or 'y' values that result from substituting the domain values into the function. For the function \( f(x) = (x+4)^2 \), since it represents a parabola that opens upwards, the vertex (the lowest point of the parabola) determines the starting point of the range. The vertex occurs when \( x = -4 \), resulting in a minimum \( y \)-value of 0. Thus, the range is

• \([0, \infty)\)

In summary, the domain describes all the possible inputs the function can handle, while the range covers all possible outputs.

One of the first steps in graphing a quadratic function like \( f(x) = (x+4)^2 \) is to create a table of values. This table helps in visualizing how the function behaves and aids in plotting the graph accurately.

**Here's how to create a table of values:**

• Choose several values of \( x \). It's usually helpful to include values that are around the vertex of the parabola. Here, we've chosen values from \(-7\) to \(0\).
• Substitute each chosen \( x \) value into the function to find the corresponding \( f(x) \) or \( y \) values. For example:
• For \( x = -7 \), \( f(-7) = (-7+4)^2 = 9\)
• For \( x = -4 \), \( f(-4) = 0 \) (this is the vertex)
• Continue calculating to fill out the rest of the table.

Once you have these ordered pairs like \((-7, 9)\) or \((-4, 0)\), you are ready to move on to the next step, plotting these points on a coordinate plane.

Once you have completed your table of values, the next step is to plot these points on a coordinate plane. This will form a visual representation of the quadratic function and help in drawing its graph.

**Steps to plot points:**

• Start by setting up your coordinate plane. Ensure the axes are labeled correctly for easy reading.
• Take each ordered pair from your table and place a point on the graph corresponding to that point. For example, for the point \((-7, 9)\), go 7 units to the left and 9 units up to place the point.
• Continue plotting other pairs like \((-6, 4)\), \((-4, 0)\), and \((0, 16)\).
• After plotting all points, draw a smooth curve that connects them. Remember, the graph of the quadratic function \( f(x) = (x+4)^2 \) will form a parabola opening upwards.

Plotting points not only illustrates how the function behaves but also unveils important aspects such as the vertex and the symmetry of the parabola, making it a crucial step in understanding quadratic functions.