Question

question 9 of 10
which of the following functions best describes this graph?
graph of a parabola opening upwards, vertex around x=-5, crossing x-axis at two points left of y-axis
a. $y = x^2 - 2x + 4$
b. $y = (x - 3)(x - 6)$
c. $y = (x + 5)(x - 4)$
d. $y = x^2 + 9x + 18$

Explanation:

Step1: Analyze the x-intercepts from the graph

The graph is a parabola opening upwards (since the coefficient of \(x^2\) will be positive) and intersects the x-axis at two points. Let's find the roots (x-intercepts) from the graph. Looking at the x-axis, the parabola crosses the x-axis at \(x = - 6\) and \(x=-3\) (wait, no, let's re - check the graph. Wait, the vertex is around \(x=-5\) area? Wait, no, let's analyze each option's roots.

For a quadratic function in factored form \(y=(x - r_1)(x - r_2)\), the roots are \(x = r_1\) and \(x=r_2\).

Let's analyze each option:

• Option A: \(y=x^{2}-2x + 4\). The discriminant \(\Delta=b^{2}-4ac=(-2)^{2}-4\times1\times4=4 - 16=- 12<0\), so it has no real roots. But the graph has real roots, so A is out.

• Option B: \(y=(x - 3)(x - 6)\). The roots are \(x = 3\) and \(x = 6\). But the graph's roots are negative (since the parabola is on the left side of the y - axis), so B is out.

• Option C: \(y=(x + 5)(x - 4)\). The roots are \(x=-5\) and \(x = 4\). One positive and one negative root. But the graph seems to have both roots negative (since the vertex is at negative x and the parabola is on the left of y - axis), so C is out.

• Option D: \(y=x^{2}+9x + 18\). Let's factor it. \(y=x^{2}+9x + 18=(x + 3)(x+6)\). The roots are \(x=-3\) and \(x=-6\). These are both negative, which matches the graph (the parabola is on the left side of the y - axis, with roots at negative x - values). Also, the parabola opens upwards (since the coefficient of \(x^{2}\) is positive), which matches the graph.

Step2: Verify the vertex or other features (optional, but to confirm)

For a quadratic function \(y = ax^{2}+bx + c\), the x - coordinate of the vertex is \(x=-\frac{b}{2a}\). For \(y=x^{2}+9x + 18\), \(a = 1\), \(b = 9\), so \(x=-\frac{9}{2}=-4.5\), which is in the negative x - region, matching the graph's vertex position.

Answer:

D. \(y = x^{2}+9x + 18\)