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A parabola is a U-shaped curve commonly seen when graphing quadratic functions. These functions can be written in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. For the given equation \( y = \frac{1}{4}x^2 \), the parabola opens upwards because the coefficient \( a = \frac{1}{4} \) is positive. Key features of a parabola include its vertex, axis of symmetry, and direction of opening. The vertex is the lowest point for upward-opening parabolas or the highest point for downward-opening ones. In our case, the vertex is at (0,0), which is both the starting point and the intercept. This makes the graph symmetric around the vertical line \( x = 0 \).Simple characteristics of a parabola include:

• Symmetrical shape around its vertex.
• Direction determined by the sign of coefficient \( a \).
• Width influenced by the absolute value of \( a \); smaller values lead to a wider parabola.

Understanding these can help in graphing and interpreting the behavior of quadratic functions.

X-intercepts are points where the graph of a function crosses the x-axis. This means where the value of \( y \) is zero. For the equation \( y = \frac{1}{4}x^2 \), the x-intercept occurs when solving \( \frac{1}{4}x^2 = 0 \). Since this simplifies to \( x^2 = 0 \), the only solution is \( x = 0 \).Thus, the x-intercept for this equation is at (0,0). This is the same point as the vertex, which is a unique property of functions where the parabola's vertex sits precisely on the x-axis.Key notes about x-intercepts:

• They show where the function changes signs (from positive to negative, or vice versa).
• In the case of this parabola, since it touches the x-axis at one point, it's also referred to as having a "single root."

Determining the x-intercepts is crucial for understanding where the parabola crosses the axis, which helps in sketching the complete graph.

The y-intercept is the point where the graph intersects the y-axis, which means \( x=0 \). For the equation we are exploring, substitute \( x = 0 \) into \( y = \frac{1}{4}x^2 \), resulting in \( y = \frac{1}{4}(0)^2 = 0 \).Therefore, the y-intercept is (0,0). This value is the initial point of the parabola when plotting the graph on the coordinate plane.Important points about y-intercepts:

• There is always exactly one y-intercept in a function if it crosses the y-axis at all.
• This point helps determine key aspects of the graph's placement within the coordinate plane.

Understanding the y-intercept provides insight into how the function begins its curve as it moves from one side to the other.

Creating a table of values is a fundamental step in graphing a quadratic equation, such as \( y = \frac{1}{4}x^2 \). By selecting specific values for \( x \), you can calculate their corresponding \( y \) values, which helps to plot the graph effectively.For this function, some chosen values might be:

• \( x = -4 \rightarrow y = 4 \)
• \( x = -2 \rightarrow y = 1 \)
• \( x = 0 \rightarrow y = 0 \)
• \( x = 2 \rightarrow y = 1 \)
• \( x = 4 \rightarrow y = 4 \)

These points assist in sketching the parabola as they outline its shape and provide data points to connect smoothly.Creating a table of values, and then plotting these on a coordinate plane, not only aids in visual understanding but also in verifying that calculations and graph placement are accurate. This method is helpful in ensuring the graph is consistent with the function's behavior.