Question

question 4 of 10
which graph is defined by the function given below?
y = (x - 5)(x - 5)
four graphs labeled a, b, c, d
click here for long description
a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

Step1: Simplify the function

First, we simplify the given function \( y=(x - 5)(x - 5) \). Using the formula \( (a - b)^2=a^2-2ab + b^2 \), here \( a=x \) and \( b = 5 \), so \( y=(x - 5)^2=x^{2}-10x + 25 \).

Step2: Find the vertex of the parabola

For a quadratic function in the form \( y=ax^{2}+bx + c \), the x - coordinate of the vertex is given by \( x=-\frac{b}{2a} \). In our function \( y=x^{2}-10x + 25 \), \( a = 1 \), \( b=- 10 \), \( c = 25 \). So \( x=-\frac{-10}{2\times1}=\frac{10}{2}=5 \).
To find the y - coordinate, substitute \( x = 5 \) into the function: \( y=(5 - 5)^2=0 \). So the vertex of the parabola is at \( (5,0) \).

Step3: Analyze the direction of the parabola

Since the coefficient of \( x^{2} \) (i.e., \( a = 1 \)) is positive, the parabola opens upwards.

Now we analyze the graphs:

• Graph A: The vertex is at \( (5,0) \) and the parabola opens upwards, which matches our findings.
• Graph B: The vertex is on the y - axis (x = 0), which does not match \( x = 5 \).
• Graph C: The vertex is not at \( (5,0) \).
• Graph D: The vertex is not at \( (5,0) \).

Answer:

A. Graph A