Problem 32 - Introductory Algebra for College Students

Question

Graph the parabola whose equation is given $$y=x^{2}-4$$

Step-by-Step Solution

Answer

The graph of the function \(y = x^2 - 4\) is a parabola opening upwards, with vertex at (0, -4) and intercepts at x = 2, x = -2 and y = -4.

Step 1: Identify the type of function and its vertex

The given function \(y = x^2 - 4\) is a quadratic function, and it is in vertex form \(y=a(x-h)^2 + k\), where (h, k) is the vertex of the parabola. In our case, the function can be rewritten as \(y=(x-0)^2 - 4\). Thus the vertex of the parabola is at the point (0, -4).

Step 2: Find the intercepts

To find the x-intercepts, we need to set y=0 in our equation and solve for x: \(0 = x^2 - 4\), and solving it gives \(x = 2\)and \(x = -2\). Thus, our x-intercepts are (2,0) and (-2,0). The y-intercept is found by setting \(x = 0\) in our equation, which gives \(y = -4\), hence, the y-intercept is at (0, -4).

Step 3: Plot the graph

Start by drawing the coordinate axes. Mark the vertex point (0, -4), the x-intercepts (2, 0) and (-2, 0), and y-intercept (0, -4) on the graph. Now, plot the parabola using these points. The parabola opens upwards as the coefficient of \(x^2\) is positive.

Key Concepts

Parabola Graphing

Graphing a parabola is an enjoyable way to visualize quadratic functions. A parabola is the u-shaped curve you get when you graph a quadratic equation like \( y = x^2 - 4 \). The most distinctive feature of a parabola is its symmetry about a straight line, known as the axis of symmetry. This specific equation, which is in the standard form \( y = ax^2 + bx + c \), simplifies to \( y = (x-0)^2 - 4 \), meaning the graph is symmetrical about the y-axis.
To graph the parabola, start by pinpointing essential points: the vertex, x-intercepts, and y-intercept. Connect these points with a smooth curved line that reflects the symmetrical nature of parabolas. Since the coefficient of \( x^2 \) is positive, this parabola opens upwards. Finding these key characteristics helps in accurately sketching the graph.

Vertex of a Parabola

The vertex of a parabola is a pivotal point on its graph. It represents the highest or lowest point, depending on the direction the parabola opens. In the equation \( y = x^2 - 4 \), the vertex form is already revealed as \( y = (x-0)^2 - 4 \), indicating that the vertex is at \( (0, -4) \).
The vertex provides crucial information:

• It tells us the minimum point of the parabola since the graph opens upwards.
• It also indicates the parabola's line of symmetry, which is the vertical line \( x=0 \) in this case.

Finding the vertex offers a foundation for plotting the rest of the parabola and understanding its direction and shape.

X-Intercepts

The x-intercepts of a parabola are the points where the graph crosses the x-axis. To find these intercepts, we set \( y=0 \) in the equation \( y = x^2 - 4 \) and solve for \( x \). This gives:

• \( 0 = x^2 - 4 \)
• \( x^2 = 4 \)
• \( x = \pm 2 \)

Therefore, the x-intercepts are at \( (2, 0) \) and \( (-2, 0) \). These points clearly show where the parabola touches or intersects the x-axis. Additionally, since x-intercepts represent real solutions to the quadratic equation, they are significant in both algebraic calculations and graphing because they help define the span of the parabola across the x-axis.

Y-Intercepts

The y-intercept of a parabola is the point where the graph meets the y-axis. This is easily found by setting \( x = 0 \) in the equation. For \( y = x^2 - 4 \), substituting gives:

• \( y = (0)^2 - 4 \)
• \( y = -4 \)

Thus, the y-intercept is at \( (0, -4) \). In terms of graphing, this point marks where the parabola crosses the y-axis, providing another vital coordinate in plotting the curve accurately. Understanding the intercepts, both x and y, contributes to a comprehensive sketch of the parabola and helps determine its interactions with the coordinate axes.