It looks like you've uploaded an image. Let me first extract and analyze its contents to help you with your query. PIL import Image

Open the uploaded image

image_path = '/mnt/data/1733497050011h0svysma.jpg' image = Image.open(image_path)

Display the image to understand its contents

image.show()I have displayed the image, but it seems I can't visually interpret it directly. I can try extracting any text or mathematical symbols from it using optical character recognition (OCR). Let me proceed with that. pytesseract

Use OCR to extract text from the image

extracted_text = pytesseract.image_to_string(image)

Display the extracted text

extracted_textThe image contains a question about the function $a(x) = -0.8(x - 5)^2 + 5$ and asks to identify the following:

1. Vertex
2. Axis of symmetry
3. Direction of the graph
4. Comparison of the width of the graph with $f(x) = x^2$

Here’s how we can approach each part of the question:

1. Vertex: The vertex form of a quadratic function is $a(x - h)^2 + k$, where $(h, k)$ is the vertex. For the given function: $a(x) = -0.8(x - 5)^2 + 5$ The vertex is $(5, 5)$.

2. Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex, so it is the equation: $x = 5$

3. Direction of the Graph: Since the coefficient of $(x - 5)^2$ is $-0.8$, which is negative, the parabola opens down.

4. Width of the Graph: The width of a parabola depends on the absolute value of the coefficient $a$ in the quadratic equation. A smaller absolute value of $a$ makes the graph wider, while a larger value makes it narrower.

   • In this case, $a = -0.8$, which is smaller in absolute value than $a = 1$ (the coefficient for the standard parabola $f(x) = x^2$).
   • Therefore, the graph is wider than the graph of $f(x) = x^2$, which corresponds to option B.

Summary:

• Vertex: $(5, 5)$
• Axis of Symmetry: $x = 5$
• Direction of the Graph: Downward
• Width of the Graph: Wider than $f(x) = x^2$ (Option B)

Would you like more details on any of these steps, or do you have further questions?

Related Questions:

1. How do you find the vertex and axis of symmetry for a general quadratic function?
2. What happens to the graph of a quadratic function when the coefficient of $x^2$ is positive versus negative?
3. How does the value of $a$ affect the "width" and "height" of the graph of a parabola?
4. What is the general form of a quadratic equation, and how does it compare to the vertex form?
5. How do you identify whether a parabola opens up or down just by looking at its equation?

Tip: When identifying the direction of a parabola, always check the sign of the coefficient in front of $(x-h)^2$. Negative values make it open downward, and positive values make it open upward.