Questions: List the intercepts and test for symmetry. y^2 = x + 121

List the intercepts and test for symmetry. y^2 = x + 121
Transcript text: List the intercepts and test for symmetry. \[ y^{2}=x+121 \]
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Solution

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Solution Steps

To find the intercepts, we need to determine where the graph intersects the x-axis and y-axis. For the x-intercept, set \( y = 0 \) and solve for \( x \). For the y-intercept, set \( x = 0 \) and solve for \( y \). To test for symmetry, check for symmetry with respect to the x-axis, y-axis, and the origin by substituting \((-x, y)\), \((x, -y)\), and \((-x, -y)\) into the equation and see if the equation remains unchanged.

Step 1: Find the Intercepts

To find the intercepts, we set \( y = 0 \) to find the x-intercept and \( x = 0 \) to find the y-intercepts.

  • X-intercept: Set \( y = 0 \) in the equation \( y^2 = x + 121 \). \[ 0^2 = x + 121 \implies x = -121 \] Thus, the x-intercept is \((-121, 0)\).

  • Y-intercepts: Set \( x = 0 \) in the equation \( y^2 = x + 121 \). \[ y^2 = 0 + 121 \implies y^2 = 121 \implies y = \pm 11 \] Thus, the y-intercepts are \((0, 11)\) and \((0, -11)\).

Step 2: Test for Symmetry

To test for symmetry, we check the equation for symmetry with respect to the x-axis, y-axis, and the origin.

  • Symmetry with respect to the x-axis: Substitute \((x, -y)\) into the equation. \[ (-y)^2 = x + 121 \implies y^2 = x + 121 \] The equation remains unchanged, indicating symmetry with respect to the x-axis.

  • Symmetry with respect to the y-axis: Substitute \((-x, y)\) into the equation. \[ y^2 = -x + 121 \] The equation changes, indicating no symmetry with respect to the y-axis.

  • Symmetry with respect to the origin: Substitute \((-x, -y)\) into the equation. \[ (-y)^2 = -x + 121 \implies y^2 = -x + 121 \] The equation changes, indicating no symmetry with respect to the origin.

Final Answer

  • X-intercept: \(\boxed{(-121, 0)}\)
  • Y-intercepts: \(\boxed{(0, 11)}\) and \(\boxed{(0, -11)}\)
  • Symmetry: \(\boxed{\text{Symmetric with respect to the x-axis}}\)
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