Question

read carefully and choose the name of the student who made the correct statement.
card 1:
mr. castillo graphed the quadratic function g(x).
alejandro says the function can be represented by g(x) = x² - 8x + 14.
ashley says the function can be represented by g(x) = x² - 5x + 10.
alejandro
ashley
justify your answer below.
your answer

Explanation:

For the multiple - choice part:

1. First, recall the vertex form of a quadratic function \(y = a(x - h)^2+k\), where \((h,k)\) is the vertex of the parabola. From the graph, we can see that the vertex of the parabola \(g(x)\) is at \((4, - 2)\). Since the parabola opens upwards, \(a = 1\).
2. Convert the vertex form to the standard form \(y=x^{2}+bx + c\). Using the vertex form \(y=(x - 4)^{2}-2\). Expand \((x - 4)^{2}-2\):

• First, expand \((x - 4)^{2}\) using the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), where \(a=x\) and \(b = 4\). So \((x - 4)^{2}=x^{2}-8x + 16\).
• Then subtract 2: \(y=x^{2}-8x + 16-2=x^{2}-8x + 14\), which matches Alejandro's function.
• For Ashley's function \(y=x^{2}-5x + 10\), the x - coordinate of the vertex is given by \(x=-\frac{b}{2a}\). Here, \(a = 1\) and \(b=-5\), so \(x=\frac{5}{2}=2.5\), which does not match the vertex x - coordinate of 4 from the graph.

Answer:

Alejandro

For the justification part: