Problem 11 - Precalculus

Question

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=-x^{2}$$

Step-by-Step Solution

Answer

The intercepts are both at (0,0), and the graph is symmetric about the origin.

Step 1: Determine the x-intercepts

To find the x-intercepts, we set the equation equal to 0: \( y = -x^2 = 0 \). Solving this equation gives us \( x^2 = 0 \), so \( x = 0 \). Thus, the x-intercept is (0,0).

Step 2: Determine the y-intercept

To find the y-intercept, we set \( x = 0 \) in the equation. Substituting gives us \( y = -(0)^2 = 0 \). Thus, the y-intercept is also (0,0).

Step 3: Determine Symmetry

The standard types of symmetry are with respect to the y-axis, the x-axis, and the origin. We check symmetry with each:- Y-axis: Substitute \( x \) with \( -x \) to check if \( f(-x) = f(x) \). Here, \( y = -(-x)^2 = -x^2 \); since \( f(-x) eq f(x) \), there is no y-axis symmetry.- X-axis: Substitute \( y \) with \( -y \). Here, \( -y = -x^2 \) or \( y = x^2 \); original and altered equations differ, so no x-axis symmetry.- Origin: Substitute \( x \) with \( -x \) and \( y \) with \( -y \). This results in \( -y = -x^2 \) which matches the original \( y = -x^2 \). Thus, the graph is symmetric about the origin.

Step 4: Graph the Equation

Now that we know the intercepts and symmetry, we can graph the equation. The graph of \( y = -x^2 \) is a downward-opening parabola with its vertex at the origin (0,0). Because there's origin symmetry, for every point \((x, y)\) on the graph, the point \((-x, -y)\) will also exist on the graph.

Key Concepts

Understanding X-Intercepts in Quadratic Functions

The x-intercepts are vital for understanding where a graph intersects the x-axis. For a quadratic function like \( y = -x^2 \), finding these intercepts involves setting the entire equation equal to zero. Essentially, you want to know where the function's output, or the y-value, is zero. In mathematical terms, we express this as \(-x^2 = 0\). Solving for \(x\), we find that \(x = 0\), making the x-intercept \((0, 0)\).

What does this mean visually on the graph? Simply put, it's the point where the graph meets the x-axis. Because quadratic functions can have up to two x-intercepts, this step is crucial. However, in this particular case, the intercept at \((0, 0)\) is the sole point where the curve touches the x-axis. Recognizing x-intercepts is not just about finding numbers; it's about understanding where the ups and downs of the parabola interact with the horizontal line of the x-axis.

Exploring Y-Intercepts in Parabolas

Y-intercepts tell us where a graph crosses the y-axis, offering a snapshot of the function when \(x = 0\). This is like taking a vertical slice of the graph. For our equation \( y = -x^2 \), finding the y-intercept means plugging 0 in for \(x\), resulting in \( y = -(0)^2 = 0 \). Thus, the y-intercept is also at \((0, 0)\).

Why is the y-intercept important? It provides a reference point from which other points can build. Every quadratic function has exactly one y-intercept, unlike potential multiple x-intercepts. In this scenario, both x and y-intercepts coincide at the origin, which isn't uncommon but not always expected. Therefore, both intercepts lying at the origin here reinforces symmetry considerations for this specific parabola.

Delving Into the Symmetry of Quadratic Functions

Symmetry in functions helps simplify graphing and analysis. Quadratic functions may exhibit different types of symmetry: with respect to the y-axis, x-axis, or the origin. Understanding these symmetries can help predict and replicate parts of the graph. Let's take a closer look using our function \( y = -x^2 \).

• Y-axis symmetry: A function is symmetric about the y-axis if \( f(x) \) equals \( f(-x) \). In this case, substituting \( -x \) results in \( y = -(-x)^2 = -x^2 \), which means this function has no y-axis symmetry since it remains unchanged.
• X-axis symmetry: Here, we check if \( y \) equals \( -y \). However, substituting results in \( y = x^2 \), differing from our original equation. This confirms no x-axis symmetry.
• Origin symmetry: A graph is symmetric about the origin if replacing \( x \) with \( -x \) and \( y \) with \( -y \) results in an equivalent expression. In this equation, \(-y = -x^2\) is equivalent to \(y = -x^2\), showing symmetry around the origin.

Understanding these symmetries helps us to recognize patterns and reduces the need for excessive calculations when graphing quadratic equations like \( y = -x^2 \), accelerating your problem-solving efficiency.