91Ó°ÊÓ

When analyzing radical functions like \( y = \sqrt{9-x} \), the domain is crucial as it tells us the set of allowable \( x \) values. For this function, the expression under the square root, \( 9-x \), must be non-negative because the square root of a negative number is not defined in the set of real numbers. This requirement gives us the inequality \( 9-x \geq 0 \), which simplifies to \( x \leq 9 \). Therefore, the domain of this function is all real numbers less than or equal to 9, or symbolically, \( (-\infty, 9] \).

The range of a radical function like this one is determined by the possible \( y \) values when \( x \) is in the domain. Since the square root function only produces non-negative outputs, the minimum value of \( y \) is 0. So, as \( x \) varies across the domain, \( y \) will also vary, producing values from 0 upward. Thus, the range of \( y = \sqrt{9-x} \) is \([0, +\infty)\). Understanding the domain and range helps us know where the graph starts, stops, and the possible values it can take.

Graphing the function \( y = \sqrt{9-x} \) helps visualize its behavior. Start by identifying key information about the function:

• The function is a square root function, typically starting higher on the axis and decreasing towards the right.
• The graph only exists for \( x \leq 9 \), as determined by the domain.

To graph, plot the main features like the intercepts and points where the graph changes direction.

• Find where the function meets the axes. On the graph of \( y = \sqrt{9-x} \), the x-intercept is (9,0) since at \( x = 9 \), \( y \) becomes 0. Meanwhile, the graph meets the y-axis at \((0, 3)\).

Create additional reference points by plugging values of \( x \) into the equation to form points such as (4.5, 2.12). These points help tailor the graph's curve accurately.

Identifying key points is a smart first step in successfully graphing \( y = \sqrt{9-x} \). These points provide a backbone that guides the sketch of the curve. Let's consider why these are essential:

• Starting Point: The function starts at \( (9, 0) \). It's a critical point as it marks where the function's value begins, suggesting the graph's smallest x-value.
• Ending Point: At \((0, 3)\), the function peaks before turning downward. Knowing this helps determine the graph's general direction.
• Mid-Points: Evaluate the function at midpoints, such as \( x = 4.5 \), where the function yields approximately 2.12. These points "anchor" the graph between the start and endpoints, ensuring the precise curve shape.

Putting these points together allows you to draw the curve confidently. By connecting the points smoothly, you authenticate the function's visual, which shows a continuous and smooth downward curve in the graph region set by the domain. Visualizing key points promotes an intuition for how the graph behaves across its domain and breathes life into its abstract formula.