Question

question 5 of 10
which of the following functions best describes this graph?
graph of a parabola opening upwards, vertex at (-1, 0), passing through (0, 1)
a. $y = (x + 1)(x + 1)$
b. $y = x^2 - 5x + 6$
c. $y = x^2 - x + 5$
d. $y = (x - 1)(x + 3)$

Explanation:

Step1: Analyze the vertex of the parabola

The graph is a parabola opening upwards with vertex at \( x=-1 \) (since it touches the x - axis at \( x = - 1\), so it has a repeated root at \( x=-1\)). A quadratic function with a repeated root at \( x = a\) can be written in the form \( y=(x - a)^2\) or \( y=(x - a)(x - a)\).

Step2: Analyze each option

• Option A: \( y=(x + 1)(x + 1)=(x + 1)^2\). The vertex of this parabola is at \( x=-1\) (since for \( y=(x - h)^2+k\), here \( h=-1,k = 0\)), and it opens upwards, which matches the graph.
• Option B: \( y=x^{2}-5x + 6=(x - 2)(x - 3)\). The roots are at \( x = 2\) and \( x=3\), and the vertex of \( y=ax^{2}+bx + c\) is at \( x=-\frac{b}{2a}=\frac{5}{2}=2.5\), which does not match the vertex at \( x=-1\) in the graph.
• Option C: \( y=x^{2}-x + 5\). The discriminant \( \Delta=b^{2}-4ac=(-1)^{2}-4\times1\times5=1 - 20=-19<0\), so it has no real roots, but the graph has a root at \( x=-1\) (touches the x - axis), so this option is incorrect.
• Option D: \( y=(x - 1)(x + 3)=x^{2}+2x - 3\). The roots are at \( x = 1\) and \( x=-3\), and the vertex is at \( x=-\frac{b}{2a}=-\frac{2}{2}=-1\), but when \( x=-1\), \( y=(-1 - 1)(-1 + 3)=(-2)\times2=-4

eq0\), so it does not touch the x - axis at \( x=-1\), it crosses the x - axis at \( x=-3\) and \( x = 1\), which does not match the graph.

Answer:

A. \( y=(x + 1)(x + 1) \)