Problem 14 - Calculus with Concepts in Calculus

Question

Determine all intercepts of the graph of the equation. Then decide whether the graph is symmetric with respect to the \(x\) axis, the \(y\) axis, or the origin. $$ y^{2}=x \sqrt{1+x^{2}} $$

Step-by-Step Solution

Answer

Intercept: (0, 0); Symmetric with respect to the x-axis.

Step 1: Finding the x-intercept

To find the x-intercepts, set \(y = 0\) in the equation \(y^2 = x\sqrt{1+x^2}\). This simplifies to \(0 = x\sqrt{1+x^2}\). For this equality to hold, \(x\) must be 0, as the square root term cannot be zero. Thus, the x-intercept is \((0, 0)\).

Step 2: Finding the y-intercepts

To find the y-intercepts, set \(x = 0\) in the equation \(y^2 = x\sqrt{1+x^2}\). This becomes \(y^2 = 0\) when \(x = 0\). Therefore, \(y = 0\). Thus, the y-intercept is also \((0, 0)\).

Step 3: Check for x-axis symmetry

To check for symmetry with respect to the x-axis, replace \(y\) in the equation with \(-y\). The equation \((-y)^2 = x\sqrt{1+x^2}\) simplifies back to \(y^2 = x\sqrt{1+x^2}\), indicating that the equation is symmetric with respect to the x-axis.

Step 4: Check for y-axis symmetry

To check for symmetry with respect to the y-axis, replace \(x\) with \(-x\) in the equation. The equation becomes \(y^2 = -x\sqrt{1+(-x)^2}\). Since the square root and the multiplication of \(-x\) changes the sign, this equation is not equivalent to the original, hence the graph is not symmetric with respect to the y-axis.

Step 5: Check for origin symmetry

To check for symmetry with respect to the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\). The equation becomes \((-y)^2 = -x\sqrt{1+(-x)^2}\), which does not simplify to the original equation \(y^2 = x\sqrt{1+x^2}\). Therefore, the graph is not symmetric with respect to the origin.

Key Concepts

x-intercept

When identifying the x-intercept of a graph, you set the y-value to zero. This is because an x-intercept is where the graph crosses the x-axis, meaning there's no vertical component (hence, y = 0). In our exercise, substituting y = 0 in the given equation:

• Equation: \(y^2 = x\sqrt{1+x^2}\)
• Substitute: \(0 = x\sqrt{1+x^2}\)

Since the right-hand side simplifies to zero for x = 0, the x-intercept is \((0, 0)\). This indicates that the graph touches or crosses the x-axis at the point (0, 0).

y-intercept

To find the y-intercept, set the value of x to zero. This is because a y-intercept indicates where the graph cuts through the y-axis, and so the horizontal component is zero (x = 0). For the exercise, substitute x = 0 into the equation:

• Equation: \(y^2 = x\sqrt{1+x^2}\)
• Substitute: \(y^2 = 0\)

Since \(y^2 = 0\) resolves to y = 0, the y-intercept is \((0, 0)\). This shows that the graph also crosses the y-axis at the origin.

graph symmetry

Graph symmetry helps you identify consistent patterns and shapes in graphs. Here, we identify symmetry by altering the equation variables. To check symmetry with respect to the x-axis, y-axis, or origin, you replace variables in the equation and simplify. In our scenario:

• X-axis Symmetry: Replace y with -y. The equation \((-y)^2 = x\sqrt{1+x^2}\) simplifies to \(y^2 = x\sqrt{1+x^2}\). This confirms the graph's x-axis symmetry.
• Y-axis Symmetry: Replace x with -x. The altered equation \(y^2 = -x\sqrt{1+(-x)^2}\) does not simplify to the original, hence no y-axis symmetry.
• Origin Symmetry: Change both x and y to -x and -y respectively. The equation \((-y)^2 = -x\sqrt{1+(-x)^2}\) does not match the original, ruling out origin symmetry.

Understanding how these transformations affect the equations is crucial in determining graph symmetry.

symmetric with respect to axes

Being symmetric with respect to the x-axis, y-axis, or the origin provides insights into the graph's shape and behavior without needing to plot numerous points. Symmetry with respect to the x-axis implies that if a point (x, y) is on the graph, then (x, -y) is also on it. This is confirmed when altering y to -y results in an equivalent equation, maintaining equality.
However, symmetry with respect to the y-axis requires that the graph remains unchanged when x is replaced with -x. If not, the presence of this symmetry is negated, as seen in our exercise result where the equation doesn't match.
Origin symmetry needs both x and y to be swapped with their negatives, having the same condition of equation equivalence. This too failed the test in the given exercise, thus not symmetric with the origin. Understanding these symmetries simplifies analyzing a graph's structure and predicting its behavior.