Question

\(y=9 x^2+7\)

Short answer

The graph of the equation \(y = 9x^2 + 7\) is a parabola with the vertex at (0,7) and opens upward.

Step-by-step solution

Identify the Type of Function

From the form of the equation \(y = 9x^2 + 7\), it can be identified as a quadratic function.

Identify the Vertex of the Parabola

The equation of the quadratic function is in the form \( y = a x ^ 2 + c \). Here, the vertex is always at the point (0,c), so in this case, the vertex of the parabola is (0,7).

Identify the Direction of the Parabola

The coefficient of \(x^2\) is positive, which means the parabola opens upwards.

Plot the Basic Points on a Graph

On a graph, plot the vertex at (0,7). The parabola opens upward, so draw the parabola with the vertex at the bottom. Some other points to include when plotting could be using x values of -1, 0, and 1. Plug these into your equation to find the corresponding y values.

Key concepts

Parabola Vertex

Understanding the vertex of a parabola is critical when studying quadratic functions, as it represents the highest or lowest point on the graph, depending on the direction of the parabola. In the given quadratic equation y = 9x^2 + 7, we can see it's not in vertex form (y = a(x-h)^2 + k), but since there is no x-term, the vertex is simply at (0,c). Therefore, the vertex of our parabola is (0,7). This point serves as a foundational reference when sketching the graph. It is also essential for determining the axis of symmetry, which in this case is the y-axis or x=0.

Remember, the vertex provides crucial information about the parabola's maximum or minimum value; in our example, this would be the minimum value since our parabola opens upward. Knowing the vertex helps anticipate the shape and location of the parabola on a coordinate plane, simplifying the process of graph plotting.

Direction of a Parabola

The direction of a parabola hinges on the coefficient of the x^2 term in a quadratic equation. In our exercise, the coefficient is positive (9 in 9x^2), which means that the parabola opens in an upward direction. This is a fundamental characteristic as it tells us that the quadratic function represents a situation with a minimum point, and there are no real x-values at which y would be negative.

To intuitively understand this concept, imagine a U-shaped curve opening towards the sky, meaning any thrown object following this path would eventually return to ground level. Conversely, if the coefficient were negative, the parabola would open downward, resembling an inverted U, akin to a waterfall cascading towards the ground.

Plotting Quadratic Graphs

The process of plotting quadratic graphs begins with the identification of the parabola's key features, including the vertex and whether it opens up or down. To plot our given function y = 9x^2 + 7, start with the vertex (0,7), which will be the lowest point of the graph. Next, establish a few more points by selecting x-values and calculating the corresponding y-values. Since symmetry plays a role, choose x-values equidistant from the vertex, like -1, 0, and 1.

After substituting these x-values into the equation, plot the resulting points on a graph. Connect these points with a smooth curve to reveal the parabolic shape. Be sure to draw the parabola continuing indefinitely in both directions, noting it's a function that has no bounds as x tends toward infinity. By following these steps, a clear visualization of the quadratic function as a parabolic graph helps enhance comprehension. Plotting graphs in this way not only brings mathematical equations to life but also solidifies understanding for real-world applications.