Question

If $f(x) = ax^2 + bx + c$ is drawn as in adjacent diagram, then $|4a + b + c|$ is equal to

• A. 2
• B. 3
• C. 5
• D. 6

Answer 1

Answer: (C)

5

Answer 2

The given function is $f(x) = ax^2 + bx + c$. From the graph, we can observe the following:

1. The y-intercept is at $(0, -3)$. Substituting $x=0$ into the function, we get $f(0) = a(0)^2 + b(0) + c = c$. Thus, $c = -3$.
2. The graph shows that the parabola intersects the line $y = -3$ at $x=0$ and at another point to the right of the y-axis. The horizontal distance between these two intersection points is given as 2. To find the x-coordinates where $f(x) = -3$, we set $ax^2 + bx + c = -3$. Substituting $c = -3$, we have $ax^2 + bx - 3 = -3$, which simplifies to $ax^2 + bx = 0$. Factoring out $x$, we get $x(ax + b) = 0$. The solutions are $x=0$ and $x = -\frac{b}{a}$. Since the horizontal distance between these two points is 2, we have $|0 - (-\frac{b}{a})| = 2$, which means $|\frac{b}{a}| = 2$. From the diagram, the other intersection point is to the right of $x=0$, so $-\frac{b}{a} = 2$, which implies $b = -2a$.
3. The vertex of the parabola is located at $x = -\frac{b}{2a}$. Substituting $b = -2a$, we get $x = -\frac{-2a}{2a} = 1$. The y-coordinate of the vertex is given as 1 unit above the line $y=-3$, so the vertex's y-coordinate is $-3 + 1 = -2$. Substituting the vertex coordinates $(1, -2)$ into the function: $f(1) = a(1)^2 + b(1) + c = a + b + c = -2$. We know $c = -3$ and $b = -2a$. Substituting these values: $a + (-2a) + (-3) = -2$ $-a - 3 = -2$ $-a = 1$ $a = -1$. Now we can find $b$: $b = -2a = -2(-1) = 2$. So, the coefficients are $a = -1$, $b = 2$, and $c = -3$. The function is $f(x) = -x^2 + 2x - 3$.

We need to find the value of $|4a + b + c|$. $4a + b + c = 4(-1) + 2 + (-3) = -4 + 2 - 3 = -5$. $|4a + b + c| = |-5| = 5$.