Question

which is the equation of the parabola?
graph of a parabola
options:
a) $y = -2x^2 - 14x + 24$
b) $y = 2x^2 + 2x + 24$
c) $y = -2x^2 - 2x + 24$

Explanation:

Step 1: Analyze the parabola's direction

The parabola opens downward, so the coefficient of \(x^2\) should be negative. This eliminates option b (\(y = 2x^2+2x + 24\)) since its \(x^2\) coefficient is positive.

Step 2: Find the vertex or use a point

Let's check the y - intercept. The y - intercept occurs when \(x = 0\). For all options, when \(x = 0\), \(y=24\), which matches the graph (since the parabola crosses the y - axis at \(y = 24\)). Now let's find the axis of symmetry. The formula for the axis of symmetry of a parabola \(y=ax^2+bx + c\) is \(x=-\frac{b}{2a}\).

For option a: \(y=-2x^2-14x + 24\), \(a=-2\), \(b=-14\). The axis of symmetry is \(x =-\frac{-14}{2\times(-2)}=\frac{14}{-4}=-3.5\)

For option c: \(y=-2x^2-2x + 24\), \(a = - 2\), \(b=-2\). The axis of symmetry is \(x=-\frac{-2}{2\times(-2)}=\frac{2}{-4}=-0.5\)

Looking at the graph, the vertex is near \(x=-0.5\) (since it's close to the y - axis on the left side). The axis of symmetry for option c is \(x=-0.5\), which matches the graph better than the axis of symmetry of option a (\(x=-3.5\)). We can also test a point. Let's take \(x = 1\).

For option a: \(y=-2(1)^2-14(1)+24=-2 - 14 + 24 = 8\)

For option c: \(y=-2(1)^2-2(1)+24=-2-2 + 24 = 20\)

From the graph, when \(x = 1\), the y - value should be negative? Wait, no, looking at the graph, when \(x = 3\) or so, the parabola crosses the x - axis, but when \(x = 1\), let's check the graph. The graph at \(x = 1\) is below the y - intercept but still positive? Wait, maybe a better point. Let's take \(x=-1\).

For option a: \(y=-2(-1)^2-14(-1)+24=-2 + 14 + 24 = 36\)

For option c: \(y=-2(-1)^2-2(-1)+24=-2 + 2 + 24 = 24\)

Wait, maybe my initial axis of symmetry approach is better. The graph's vertex is closer to \(x = - 0.5\), so option c is correct.

Answer:

c) \(y=-2x^2 - 2x + 24\)