Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the equation. \(y=2-x^{2}\)

Step-by-Step Solution

Answer

Our graph is a downward opening parabola with its vertex at (0,2), and intersects the x-axis at \(\sqrt{2}\) and \(-\sqrt{2}\)

Step 1: Find the y intercept

To find the y intercept, put x=0 in the given equation. So, here \(y = 2 - (0)^{2} = 2\). So, the y-intercept for the equation is at point (0, 2). This is also the vertex of the parabola.

Step 2: Find the roots of the equation

The roots of the equation are the points where y = 0. To find those, we set y=0 and solve the equation for x: \(0 = 2 - x^{2}\). Solving this equation we get two roots: \(x = \sqrt{2}\) and \(x = -\sqrt{2}\)

Step 3: Sketch the graph

First, point the y-intercept on the graph at the point (0, 2). Then, mark two more points on x-axis, one at \(\sqrt{2}\) and the other at \(-\sqrt{2}\). Since we know the parabola opens downward (negative x^{2} term), the sketch will look like a 'U' inverted with its vertex at (0,2).

Key Concepts

Y-Intercept

Understanding the y-intercept of a quadratic equation is crucial for graphing the parabola it represents. The y-intercept is the point where the graph crosses the y-axis. Mathematically speaking, it is found by setting the x-variable to zero and solving for y. In the given equation, y=2-x^2, the calculation for the y-intercept would be y = 2 - (0)^2 = 2. Hence, the y-intercept (0, 2) becomes the starting point for drawing the curve on the graph. It is worthwhile to note that the y-intercept can provide immediate insight into the vertical position of the parabola on the graph.

When sketching, plot the y-intercept first. This point serves as a key reference and, as we have discovered in our exercise, coincides with the vertex of the parabola when the equation is in standard form and the coefficient of x^2 is negative.

Roots of the Equation

The roots of a quadratic equation, also known as its zeros or x-intercepts, are the values of x where the graph intersects the x-axis. Essentially, these are the solutions to the equation when set equal to zero. To find the roots of the equation y = 2 - x^2, we set y to 0, yielding 0 = 2 - x^2. Solving for x, we obtain the roots x = \( \sqrt{2} \) and x = -\( \sqrt{2} \). Plotting these points on the graph helps in shaping the curve of the parabola.

It's important to recognize that a parabola can have zero, one, or two real roots depending on where it sits in relation to the x-axis. The roots provide significant information about the symmetry and width of the parabola, and they are essential in determining the overall shape of the graph.

Vertex of a Parabola

The vertex of a parabola is arguably the most defining feature of its graph. It represents the highest or lowest point, depending on whether the parabola opens upwards or downwards. For the equation y=2-x^2, the vertex is at the y-intercept, which is the point (0, 2). This happens because the coefficient of x^2 is negative, indicating the parabola opens downwards, and the squared term is in its simplest form, which places the vertex along the y-axis.

To further understand the vertex, consider its role in the equation's axis of symmetry. This is the vertical line that runs through the vertex and divides the parabola into two mirror-image halves. For our equation, the axis of symmetry is x = 0. Knowing the vertex and the axis of symmetry simplifies the graphing process, providing a clear picture of the parabola's direction and offering a foundation for plotting additional points on the graph.