Problem 26 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ y=x^{2}+2 $$

Step-by-Step Solution

Answer

The graph is a parabola opening upwards with a y-intercept at (0, 2) and no x-intercepts. It is symmetric about the y-axis.

Step 1: Understanding the Equation

The given equation is a parabola in standard form: \( y = x^2 + 2 \). This means it is a quadratic equation, and its graph is a parabola opening upwards.

Step 2: Making a Table of Values

Choose values for \( x \) and calculate corresponding \( y \) values. Let's select \( x = -2, -1, 0, 1, 2 \):- When \( x = -2 \), \( y = (-2)^2 + 2 = 4 + 2 = 6 \)- When \( x = -1 \), \( y = (-1)^2 + 2 = 1 + 2 = 3 \)- When \( x = 0 \), \( y = 0^2 + 2 = 2 \)- When \( x = 1 \), \( y = 1^2 + 2 = 1 + 2 = 3 \)- When \( x = 2 \), \( y = 2^2 + 2 = 4 + 2 = 6 \)So, the table of values is:\[\begin{array}{c|c}x & y \hline-2 & 6 \-1 & 3 \0 & 2 \1 & 3 \2 & 6 \\end{array}\]

Step 3: Sketching the Graph

Use the table of values to plot the points on a coordinate plane: \((-2, 6), (-1, 3), (0, 2), (1, 3), (2, 6)\). Connect these points with a smooth curve representing a parabola that opens upwards.

Step 4: Finding the Intercepts

**x-intercept:** Set \( y = 0 \) in the equation:\[ 0 = x^2 + 2 \]This equation, \( x^2 = -2 \), has no real solutions because a square can't be negative. So, there are no x-intercepts.**y-intercept:** Set \( x = 0 \) in the equation and solve for \( y \):\[ y = 0^2 + 2 = 2 \]The y-intercept is \((0, 2)\).

Step 5: Testing for Symmetry

A parabola of the form \( y = x^2 + c \) is symmetric with respect to the y-axis. To verify, substitute \( x \) with \( -x \) in the equation:\[ y = (-x)^2 + 2 = x^2 + 2 \]Since the equation remains unchanged, the graph is symmetric with respect to the y-axis.

Key Concepts

Parabola

A parabola is a smooth, symmetrical curve that corresponds to a quadratic equation. In this context, the quadratic equation given is in the standard format: \( y = x^2 + 2 \). This shows that the parabola will open upwards. The term \( x^2 \) indicates that the graph is a perfect symmetrical shape centered around the y-axis. The \( + 2 \) shifts the entire parabola upwards by 2 units on the y-axis, which means the vertex of this standard parabola isn't at the origin but at the point \( (0, 2) \). Parabolas are a key concept of quadratic functions and their interaction with axes is crucial in understanding their behavior visually.

x-intercept

In mathematics, an x-intercept is a point where a graph crosses the x-axis. At this point, the value of \( y \) is zero. To find the x-intercepts of a quadratic function, set the equation to zero and solve for \( x \). For the equation \( y = x^2 + 2 \), you'll set: \[ 0 = x^2 + 2 \] Therefore, solving \( x^2 + 2 = 0 \) involves finding \( x^2 = -2 \). This results in no real solution since the square of a real number is always non-negative. Hence, this parabola does not intersect the x-axis, meaning it has no x-intercepts. This implies that the parabola lies completely above the x-axis as it opens upwards from the vertex \( (0, 2) \).

y-intercept

A y-intercept is a point where the graph crosses the y-axis, meaning the value of \( x \) is zero. To determine the y-intercept of a quadratic function, substitute \( x = 0 \) into the equation and solve for \( y \). For the function \( y = x^2 + 2 \), substituting \( x = 0 \) results in: \[ y = 0^2 + 2 = 2 \] This calculation tells us that the parabola intersects the y-axis at the point \( (0, 2) \). The y-intercept is often the starting point for graphing a quadratic equation as it provides a firm anchor on the graph.

Symmetry

Symmetry in a graph means that one side is a mirror reflection of the other. For parabolas, symmetry is significant because it indicates predictability in the graph. A quadratic equation of the form \( y = x^2 + c \), like in our example, is always symmetric about the y-axis. This is because if you substitute \( x \) with \( -x \), the equation remains unchanged: \( y = (-x)^2 + 2 = x^2 + 2 \). This symmetry indicates that the parabola's left side is a mirror of its right side, both centered around the y-axis. Recognizing this symmetry allows students to predict the shape and path of the graph even before plotting every point.

Graphing

Graphing is the process of visually depicting a function or an equation on a coordinate system. To graph the function \( y = x^2 + 2 \), you can start by calculating specific points using a table of values. As solved earlier, such points include \([(-2, 6), (-1, 3), (0, 2), (1, 3), (2, 6)]\). Plot these points on the coordinate plane, and draw a smooth curve that connects them to visualize the graph of the parabola.
When graphing parabolas, consider key attributes such as the vertex, direction of opening, y-intercept, and symmetry. These attributes assist in ensuring the graph accurately represents the quadratic function. The graph should show a smooth and continuous curve that matches symmetric properties, giving students a complete picture of the parabolic shape.