Problem 27 - 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. $$ 4 y=x^{2} $$

Step-by-Step Solution

Answer

The parabola is symmetric about the y-axis with intercepts (0,0).

Step 1: Rewrite the Equation

The equation given is \( 4y = x^2 \). First, solve for \( y \) to get it into the form \( y = f(x) \). Divide each side by 4 to get \( y = \frac{x^2}{4} \). This is now in the standard form for a quadratic function, making it easier to analyze.

Step 2: Make a Table of Values

Choose a set of \( x \)-values and calculate corresponding \( y \)-values using the equation \( y = \frac{x^2}{4} \). For example:- If \( x = -2 \), then \( y = \frac{(-2)^2}{4} = 1 \).- If \( x = 0 \), then \( y = \frac{0^2}{4} = 0 \).- If \( x = 2 \), then \( y = \frac{2^2}{4} = 1 \).- If \( x = 4 \), then \( y = \frac{4^2}{4} = 4 \).This gives us the set of points: \((-2, 1), (0, 0), (2, 1), (4, 4)\).

Step 3: Sketch the Graph

Plot the points from the table on a graph:- \((-2, 1)\)- \((0, 0)\)- \((2, 1)\)- \((4, 4)\)Connect these points to form a parabola. The graph should be a symmetrical U-shape opening upwards with the vertex at the origin \((0, 0)\).

Step 4: Find the Intercepts

**X-intercepts:** These occur where \( y = 0 \). From \( 4y = x^2 \), setting \( y = 0 \) gives \( x^2 = 0 \). So, \( x = 0 \) is the x-intercept.**Y-intercept:** This occurs where \( x = 0 \). Substituting \( x = 0 \) into \( y = \frac{x^2}{4} \) gives \( y = 0 \). Thus, the y-intercept is also \( 0 \).

Step 5: Test for Symmetry

To check for symmetry:- **Y-axis symmetry:** Replace \( x \) with \( -x \) in the equation \( y = \frac{x^2}{4} \). It becomes \( y = \frac{(-x)^2}{4} = \frac{x^2}{4} \), which is identical to the original equation. Hence, it's symmetric about the y-axis.- **X-axis symmetry:** Not applicable to this quadratic.- **Origin symmetry:** For symmetry about the origin, \( y \) should equal \( -y \) and \( x \) with \( -x \), which is not the case here, so the graph is not symmetric about the origin.

Key Concepts

Intercepts

Intercepts are points where a graph crosses the x-axis or y-axis. They are fundamental in understanding the behavior of quadratic functions and serve as valuable checkpoints in graphing these equations.

• X-intercepts: These are found by setting the y-value to zero and solving for x. For the equation given, \( 4y = x^2 \), the x-intercept occurs when \( y = 0 \). Substituting \( y = 0 \) results in \( x^2 = 0 \), giving \( x = 0 \). Thus, the graph crosses the x-axis at the origin.


• Y-intercepts: These occur when x is zero. By substituting \( x = 0 \) in the equation \( y = \frac{x^2}{4} \), we resolve \( y = 0 \). This means the y-intercept is also at the origin, the same point as the x-intercept.

Recognizing intercepts not only helps in starting your graph but can also verify the correctness of plotted points and the overall shape of the graph. In this scenario, the intercepts reinforce that the graph passes through the origin (0, 0).

Symmetry in Graphs

For quadratic functions, examining symmetry provides insights into the shape and orientation of the graph. Symmetry simplifies graphing by reducing the amount of calculation needed and helps in predicting unseen parts of the graph.

• Y-axis Symmetry: For the equation \( y = \frac{x^2}{4} \), if replacing \( x \) with \( -x \) yields the same equation, the graph is symmetric about the y-axis. Calculating \( y = \frac{(-x)^2}{4} = \frac{x^2}{4} \) confirms this symmetry, indicating that the left side of the graph mirrors the right side.


• X-axis and Origin Symmetry: X-axis symmetry is less common with quadratic forms and requires that replacing \( y \) with \( -y \) yields the original equation, which is not the case here. Similarly, origin symmetry involves transforming both x and y, which doesn't hold for our equation either. Therefore, this graph lacks both x-axis and origin symmetry.

Understanding these symmetry properties helps in forming a mental image of the quadratic's graph, ensuring accuracy when sketching.

Parabolic Graphs

Parabolic graphs are a hallmark of quadratic equations, displaying unique characteristics that make them distinct and recognizable. The graph of a quadratic function is called a parabola and typically takes a U-shape.

• Shape and Direction: The equation \( y = \frac{x^2}{4} \) produces a parabola that opens upwards. This is due to the positive coefficient of \( x^2 \). If the coefficient were negative, the parabola would open downwards.


• Vertex: The vertex is a pivotal point on the parabola. Here, it is at \( (0, 0) \), serving as the minimum point of the graph since the parabola opens upwards. The vertex acts as the axis of symmetry, dividing the graph into two mirror-image halves.


• Graph Behavior: As \( x \) becomes very large or very small, the y-values increase, indicating the arms of the parabola extend indefinitely upwards. The nature of parabolas makes them widely applicable in physics and engineering, everywhere from projectile paths to satellite dishes.

Embracing the simple yet symmetrical beauty of parabolic graphs makes graphing quadratic functions an intuitive task.