Graph the following function on the axes provided. f(x) = -5.2 – 21 2 – 9 for x < -3=”” for=”” x=””> 4 Click and drag to make a line. Click the line to delete it. Click on an endpoint of a line to change it. y 10 9 8 7 6 5 4 3 1 2 8 -6 -5 – 1 w 5 6 7 9 10 -1 -5 -6 -7 -10
The Correct Answer and Explanation is:
To graph the function correctly and explain it, we first need to clarify the piecewise function, which seems garbled in your question. Based on your input, it appears to be a piecewise function defined as: f(x)={−5.2,for x<−3−21×2−9,for x>4f(x) = \begin{cases} -5.2, & \text{for } x < -3 \\ -21x^2 – 9, & \text{for } x > 4 \end{cases}
Let’s break this down:
1. Graphing f(x)=−5.2f(x) = -5.2 for x<−3x < -3:
This part is a horizontal line at y=−5.2y = -5.2, and it only exists for values where x<−3x < -3. So you would draw a straight line horizontally at y=−5.2y = -5.2, extending leftward from x=−3x = -3. The point at x=−3x = -3 is not included, so place an open circle at (−3,−5.2)(-3, -5.2).
2. Graphing f(x)=−21×2−9f(x) = -21x^2 – 9 for x>4x > 4:
This part is a quadratic equation (a parabola). Since the coefficient of x2x^2 is negative, the parabola opens downward. But we only graph the part for x>4x > 4.
Let’s calculate some points for x>4x > 4:
- When x=5x = 5:
f(5)=−21(25)−9=−525−9=−534f(5) = -21(25) – 9 = -525 – 9 = -534 - When x=4.1x = 4.1:
f(4.1)=−21(16.81)−9≈−353.01−9≈−362.01f(4.1) = -21(16.81) – 9 ≈ -353.01 – 9 ≈ -362.01
You’ll notice the values quickly drop to large negative numbers. So the graph steeply curves downward to the right of x=4x = 4. Start the parabola right after x=4x = 4 with an open circle at x=4x = 4, then plot a curved line downward (left to right).
Important Notes for Graphing:
- Use open circles at x=−3x = -3 and x=4x = 4 to show that those points are not included.
- The horizontal line from x=−3x = -3 going left is flat at y=−5.2y = -5.2.
- The parabola segment begins just after x=4x = 4 and curves downward steeply.
Explanation Summary (300+ Words):
This is a piecewise-defined function, meaning it behaves differently based on the value of xx. Understanding how to graph this involves analyzing each “piece” separately and applying correct graphing principles.
The first piece of the function is a constant function f(x)=−5.2f(x) = -5.2 defined for all values where x<−3x < -3. This tells us that no matter what value of xx we choose (as long as it’s less than -3), the value of the function f(x)f(x) is always -5.2. When graphing this, we draw a horizontal line at y=−5.2y = -5.2 extending to the left. Since the function only holds for x<−3x < -3, and not at x=−3x = -3, we indicate this with an open circle at the point (−3,−5.2)(-3, -5.2), showing that this endpoint is not included.
The second piece of the function is a quadratic expression f(x)=−21×2−9f(x) = -21x^2 – 9, defined for values where x>4x > 4. This function represents a parabola opening downward due to the negative coefficient of x2x^2. However, the domain restriction of x>4x > 4 means we only draw the part of the parabola to the right of x=4x = 4. We again use an open circle at x=4x = 4 to show this point is not included in the graph. The values of f(x)f(x) in this region drop very rapidly, and the graph curves downward steeply as xx increases.
To summarize, this function is a combination of a flat line (left side) and a steep, downward-opening curve (right side), with a break in the graph between x=−3x = -3 and x=4x = 4, where no function is defined. Understanding such behavior is essential in graphing piecewise functions accurately.