Problem 163 - Elementary Algebra

Question

Recognize the Graph of a Quadratic Equation in Two Variables. $$ y=x^{2}+3 $$

Step-by-Step Solution

Answer

The graph is a parabola opening upwards with vertex at (0, 3) and axis of symmetry x = 0.

Step 1: Understand the Basic Form

A quadratic equation in two variables typically has the form \(y = ax^2 + bx + c\). The given equation is \(y = x^2 + 3\). Here, \(a = 1\), \(b = 0\), and \(c = 3\). The absence of the \(bx\) term and the positive \(a\) value are important characteristics.

Step 2: Analyze the Coefficients

In the equation \(y = x^2 + 3\), the coefficient of \(x^2\) is positive (\(a = 1\)). This information tells us that the parabola opens upwards.

Step 3: Identify the Vertex

The vertex form of a quadratic equation is \(y = a(x - h)^2 + k\). Comparing \(y = x^2 + 3\) with this form, we see that \(h = 0\) and \(k = 3\). Therefore, the vertex of the parabola is at \((0, 3)\).

Step 4: Determine the Axis of Symmetry

The axis of symmetry for a parabola given by \(y = ax^2 + bx + c\) is \(x = -\frac{b}{2a}\). Here, since \(b = 0\), the axis of symmetry is \(x = 0\), which is the y-axis.

Step 5: Sketch the Graph

To sketch the graph, start at the vertex \((0, 3)\). Since the parabola opens upwards and the axis of symmetry is the y-axis, plot points on either side of the vertex. For example, if \(x=1\), \(y=1^2 + 3 = 4\). Similarly, for \(x=-1\), \(y=(-1)^2 + 3 = 4\). Plot these points and draw a smooth curve through them, ensuring the curve is symmetric around the y-axis.

Key Concepts

Parabola

A parabola is a symmetric curve formed by the graph of a quadratic equation in the form of \(y = ax^2 + bx + c\). The given equation is \(y = x^2 + 3\). Parabolas can open upwards or downwards based on the coefficient of \(x^2\). Here, the coefficient of \(x^2\) is 1 (positive), so the parabola opens upwards. Parabolas have unique features like the vertex, the axis of symmetry, and the direction they open.
Understanding these features helps in sketching the graph effectively.

Vertex

The vertex is the highest or lowest point on the parabola. For the equation \(y = x^2 + 3\), it is the lowest point since the parabola opens upwards. The vertex form of a quadratic equation is \(y = a(x - h)^2 + k\). Here, \(h = 0\) and \(k = 3\), so the vertex is at \((0, 3)\).
The vertex is the point where the parabola changes direction. It's crucial for drawing an accurate graph and understanding the parabola's shape.

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex of the parabola, dividing it into two mirror-image halves. For the equation \(y = x^2 + 3\), the axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\). Since \(b = 0\) in this case, the axis of symmetry is \(x = 0\), which is simply the y-axis.
This axis helps in plotting the graph as any point on one side of the axis has a corresponding point on the opposite side.

Upward Parabola

An upward parabola opens in the positive y-direction. This happens when the coefficient of \(x^2\) (the \(a\) value) is positive. In the equation \(y = x^2 + 3\), \(a = 1\) which is positive, indicating the parabola opens upward.
The upward direction means the vertex is the minimum point on the graph. Points on either side of the vertex will be higher.

Coefficient Analysis

Coefficients in a quadratic equation \(y = ax^2 + bx + c\) define key attributes of the parabola:

• \textsuperscript{th}e \(a\) coefficient determines direction (upward/downward) and width.
• \textsuperscript{th}e \(b\) coefficient affects the skew and position relative to the y-axis.
• \textsuperscript{th}e \(c\) coefficient determines the y-intercept.

In the given equation \(y = x^2 + 3\), \(a = 1\) (positive, opens upward), \(b = 0\) (symmetric), and \(c = 3\) (y-intercept at (0, 3)).