Question

Araceli is drawing a picture on a square piece of paper that will be placed in a rectangular frame with a mat around the picture. The width of the frame will be 3 inches more than the width of the picture, and the length of the frame will be 5 inches more than the length of the picture. The are of the picture and frame together is given by the function: A(x) = (x+3)(x+5)

Find the standard form of the function A(x) and the y-intercept. Interpret the y-intercept in the context.

Answer 1

The standard form of the function A(x) is A(x) = x² + 8x + 15, and the y-intercept is 15, which represents the area of the mat when the width of the picture is zero.

Step-by-step explanation:

To find the standard form of the function A(x) = (x+3)(x+5), we need to expand the brackets using the distributive property. By multiplying each term in one parenthesis by each term in the other parenthesis, we get:

A(x) = x² + 5x + 3x + 15

Combine like terms:

A(x) = x² + 8x + 15

This is the standard form of the function A(x), which is a quadratic equation representing the area of the picture and frame together.

The y-intercept is found by evaluating A(x) when x = 0:

A(0) = 0² + 8(0) + 15 = 15

So, the y-intercept of the function A(x) is 15. In the context of this problem, the y-intercept represents the area of the frame when the width of the picture is zero, which could be interpreted as the area of the mat alone.

Answer 2

The standard form of the function A(x) is x² + 8x + 15, and the y-intercept is 15, which represents the area of the frame's borders alone.

Step-by-step explanation:

To find the standard form of the function A(x) = (x+3)(x+5), we need to expand the binomials:

A(x) = x² + 5x + 3x + 15

A(x) = x² + 8x + 15

Therefore, the standard form of the area function is A(x) = x² + 8x + 15. The y-intercept in the context of this problem is the point at which the line of the graph representing the area of the picture and frame crosses the y-axis. Since the value of x is zero at the y-intercept, we find the area at the y-intercept by evaluating A(0), which gives us A(0) = (0+3)(0+5) = 15. This y-intercept value, 15, represents the area of the frame and picture when the picture has zero width and length, essentially pointing to the area contributed by the frame's borders alone.