Graph the parabola.
$y=2 x^2-20 x+55$
The given equation is \(y = 2x^2 - 20x + 55\). This is a quadratic function of the form \(y = ax^2 + bx + c\), with coefficients \(a = 2\), \(b = -20\), and \(c = 55\). Since \(a = 2 > 0\), the parabola opens upwards. To analyze its properties, we want to convert it to the vertex form \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
To complete the square, we factor out the coefficient of \(x^2\) from the first two terms:
\[y = 2(x^2 - 10x) + 55\]
We then take half of the coefficient of \(x\) (which is \(-10/2 = -5\)), square it (\((-5)^2 = 25\)), and add and subtract this value inside the parentheses:
\[y = 2(x^2 - 10x + 25 - 25) + 55\]
\[y = 2((x - 5)^2 - 25) + 55\]
\[y = 2(x - 5)^2 - 50 + 55\]
\[y = 2(x - 5)^2 + 5\]
The vertex is \((h, k) = (5, 5)\). The axis of symmetry is the line \(x = 5\).
The y-intercept occurs where \(x = 0\):
\[y = 2(0)^2 - 20(0) + 55 = 55\]
So, the y-intercept is \((0, 55)\). To find the x-intercepts, we set \(y = 0\):
\[2(x - 5)^2 + 5 = 0\]
\[2(x - 5)^2 = -5\]
\[(x - 5)^2 = -2.5\]
Since the square of any real number cannot be negative, this equation has no real solutions. Thus, the parabola does not cross the x-axis.
We can calculate a few additional points to help visualize the parabola:
The parabola has its lowest point at \((5, 5)\), extends upwards, passes through \((4, 7)\) and \((6, 7)\), and intersects the y-axis at \((0, 55)\).
Understanding the properties of a parabola is essential in physics and engineering. For example, when an object is thrown into the air, its height over time follows a parabolic path. By identifying the vertex, you can determine the maximum or minimum height the object reaches, which is critical for safety calculations in construction or ballistics. Learning to graph these functions allows you to visualize these real-world motions accurately and predict where an object will be at any given moment.