+523 The parabola $y = ax^2 + bx + c$ is graphed below. Find $a \cdot b \cdot c.$ (The grid lines are one unit apart.)
+7 $a=\frac{2}{3} , b=-4,$ and $c=7$. That means that the answer is $\frac{2}{3} \cdot -4 \cdot 7$=-56/3$
+1804 We need to note a couple of important points to find the equation of this parabola.
We need one random point along the parabola and the vertex.
Let's choose the point (0, 7) since it's the y intercept, and the vertex of the graph is \((3, 1)\)
Now, since we have the vertex of the graph, we can put this equation into intercept form.
We have
\( f(x) = m ( x-3)^2 + 1\) where m is a real number.
Now, we simply subsitute in the point we retrireved, (0, 7), and plug it in the find m.
We have that
\(7 = m(0-3)^2+1\\ 7 = 9m+1\\ m = 2/3\)
Now that we have m, we have the full equation of the parabola in INTERCEPT form.
We need to convert it into standard form to find a,b,c.
The equation is now
\( f(x) = 2/3(x-3)^2 + 1 \)
This expands to the equation
\(2/3 x^2 -4 + 7 \) meaning that we found each value of a,b, and c.
We have that \(a = 2/3 , b = - 4 , c = 7 \). Multiplying these 3 together, we get \(2/3*(-4)*7 = -28 *2/3 = -56/3\)
So -56/3 is our final answer.
Thanks! :)
