Question

question 2 of 10
which of the following functions best describes this graph?
graph of a parabola opening upwards, with vertex in the fourth quadrant and x-intercepts positive
click here for long description
a. $y = x^2 - 9x + 18$
b. $y = x^2 - 5x + 6$
c. $y = (x + 3)(x + 6)$
d. $y = (x - 3)(x + 4)$

Explanation:

Step1: Find x-intercepts from graph

The graph is a parabola opening upwards (since coefficient of \(x^2\) is positive for all options) and crosses the x - axis at two positive x - values (from the graph, the roots are positive).

Step2: Find roots of each option

- For option A: \(y=x^{2}-9x + 18\). Factor it: \(x^{2}-9x + 18=(x - 3)(x - 6)\). Roots are \(x = 3\) and \(x=6\) (both positive).
- For option B: \(y=x^{2}-5x + 6\). Factor it: \(x^{2}-5x + 6=(x - 2)(x - 3)\). Roots are \(x = 2\) and \(x = 3\) (both positive). But let's check the y - intercept. The y - intercept of a quadratic \(y=ax^{2}+bx + c\) is \(c\). For the graph, when \(x = 0\), \(y\) is positive (close to 10). For option B, when \(x = 0\), \(y=6\). For option A, when \(x = 0\), \(y = 18\) which is closer to the graph's y - intercept (around 10 - 12? Wait, no, wait the graph: when \(x = 0\), the y - value is positive, let's re - evaluate. Wait, maybe I made a mistake. Wait, let's check the vertex. The x - coordinate of the vertex of \(y=ax^{2}+bx + c\) is \(x=-\frac{b}{2a}\).
- For option A: \(a = 1\), \(b=-9\), vertex \(x=\frac{9}{2}=4.5\)
- For option B: \(a = 1\), \(b=-5\), vertex \(x=\frac{5}{2}=2.5\)
- For option C: \(y=(x + 3)(x + 6)=x^{2}+9x+18\). Roots are \(x=-3\) and \(x=-6\) (negative, so eliminate as graph has positive roots)
- For option D: \(y=(x - 3)(x + 4)=x^{2}+x - 12\). Roots are \(x = 3\) and \(x=-4\) (one negative, eliminate as graph has two positive roots)

Now, between option A and B. The graph's vertex: the x - coordinate of the vertex. For option A, vertex at \(x = 4.5\), for option B, vertex at \(x = 2.5\). From the graph, the vertex is between \(x = 3\) and \(x = 6\)? Wait, the graph shows that the vertex is at a higher x - value. Wait, maybe I misread the graph. Wait, the original graph: when we look at the x - axis, the roots are around, say, 3 and 6? Wait, no, let's re - factor:

Wait, option A: \(y=(x - 3)(x - 6)\), roots at 3 and 6. Option B: \(y=(x - 2)(x - 3)\), roots at 2 and 3. The graph in the problem: the two x - intercepts are at positive x, and the vertex is between them. But the y - intercept: when \(x = 0\), for option A, \(y = 18\), for option B, \(y = 6\). The graph's y - intercept is positive, and from the graph, when \(x = 0\), \(y\) is greater than 6. Wait, but maybe I made a mistake. Wait, let's check the options again. Wait, option B: \(y=x^{2}-5x + 6\), let's check the vertex. The vertex of \(y=ax^{2}+bx + c\) is at \((-\frac{b}{2a},y(-\frac{b}{2a}))\). For option B, \(a = 1\), \(b=-5\), so \(x=\frac{5}{2}=2.5\), \(y=(2.5)^{2}-5\times2.5 + 6=6.25-12.5 + 6=-0.25\). For option A, \(x=\frac{9}{2}=4.5\), \(y=(4.5)^{2}-9\times4.5 + 18=20.25-40.5 + 18=-2.25\). Wait, the graph's vertex is above the x - axis? No, the graph shows a vertex below the x - axis? Wait, no, the graph: the parabola opens upwards, and the vertex is below the x - axis? Wait, no, the graph in the picture: the parabola is above the x - axis? Wait, no, the y - axis: the graph starts at the top (y - axis, positive y), then goes down, touches the x - axis at two points, then goes up. Wait, no, the graph is a parabola opening upwards, with two x - intercepts (crossing the x - axis), and the vertex is below the x - axis? Wait, no, if it crosses the x - axis at two points, the vertex is between them and below the x - axis.

Wait, let's re - analyze the roots:

Option A: roots at 3 and 6 (both positive)

Option B: roots at 2 and 3 (both positive)

Option C: roots at - 3 and - 6 (negative, eliminate)

Option D: roots at 3 and - 4 (one negative, eliminate)

Now, the y - intercept: for option A, when \(x = 0\), \(y = 18\); for option B, \(y = 6\). The graph's y - intercept (when \(x = 0\)) is around, looking at the graph, between 10 and 20? Wait, the graph's y - axis has a mark at 10, and the graph starts above 10? Wait, no, the graph: the left side is at x = 0, y is around 10 - 12. So option A has y - intercept 18, which is higher, option B has 6, which is lower. Wait, maybe I made a mistake in the vertex. Wait, no, the key is the roots. The graph's roots are two positive numbers, and from the options, option B has roots 2 and 3, option A has 3 and 6. Wait, the graph's x - intercepts: looking at the x - axis, the distance between the two roots. In option B, the roots are 2 and 3 (distance 1), in option A, 3 and 6 (distance 3). The graph's roots seem to be further apart? Wait, maybe the correct answer is B? Wait, no, wait let's check the factoring again.

Wait, option B: \(y=x^{2}-5x + 6=(x - 2)(x - 3)\), roots at 2 and 3.

Option A: \(y=x^{2}-9x + 18=(x - 3)(x - 6)\), roots at 3 and 6.

The graph: the two x - intercepts are at positive x, and the vertex is between them. Now, the y - intercept: when \(x = 0\), the graph has a positive y - value. For option B, \(y(0)=6\), for option A, \(y(0)=18\). The graph's y - intercept is closer to 10? Wait, maybe the original graph's y - intercept is around 6 - 18? Wait, maybe I made a mistake. Wait, let's check the options again. Wait, the graph: the parabola is opening upwards, with two x - intercepts (positive x), so options A and B are candidates. Now, let's check the value of the function at \(x = 4\).

For option A: \(y=16-36 + 18=-2\)

For option B: \(y=16-20 + 6=2\)

Wait, the graph at \(x = 4\): is it positive or negative? The graph: when x is between the two roots, the function is negative (since it opens upwards). So at \(x = 4\), which is between 3 and 6 (option A's roots) or between 2 and 3 (option B's roots). If \(x = 4\) is between 3 and 6, then option A's function is negative there, which matches the graph (vertex below x - axis). Option B's roots are 2 and 3, so at \(x = 4\), which is outside the roots, the function is positive, which does not match the graph (since the graph at \(x = 4\) should be between the roots and negative). So option A: roots at 3 and 6, so at \(x = 4\) (between 3 and 6), the function is negative, which matches the graph. Option B: roots at 2 and 3, so at \(x = 4\) (outside the roots), the function is positive, which does not match. So option A is correct? Wait, but earlier when I calculated the vertex of option A, \(y=-2.25\), which is below the x - axis, and option B's vertex \(y=-0.25\), also below. Wait, maybe I made a mistake in the y - intercept. Wait, the graph's y - intercept: when \(x = 0\), the graph is at y = 18? No, the graph's y - axis: the top of the graph on the y - axis is around 10, but maybe the scale is different. Wait, the x - axis has marks from - 10 to 10, with each small tick maybe 1 unit. The graph starts at (0, y) where y is around 10 - 12, then goes down, crosses the x - axis at two positive points, then goes up.

Wait, let's check the options again. Option A: \(y=x^{2}-9x + 18\), when \(x = 0\), \(y = 18\); option B: \(y = 6\). If the graph's y - intercept is around 10 - 18, then option A is better. So the correct answer is A.

Answer:

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