How to test for symmetry of a graph with respect to the y-axis

📚 Quick Study Guide

• 🔍 Y-Axis Symmetry Definition: A graph is symmetric with respect to the y-axis if, for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph. In simpler terms, the graph looks the same on both sides of the y-axis.
• 🧪 Testing Algebraically: To test for y-axis symmetry algebraically, replace $x$ with $-x$ in the equation of the graph. If the resulting equation is equivalent to the original equation, then the graph is symmetric with respect to the y-axis.
• 💡 Example: Consider the equation $y = x^2$. Replacing $x$ with $-x$ gives $y = (-x)^2 = x^2$, which is the original equation. Therefore, the graph of $y = x^2$ is symmetric with respect to the y-axis.
• 📝 Key Formula: The test is to verify if $f(x) = f(-x)$.

Practice Quiz

1. Which of the following equations represents a graph that is symmetric with respect to the y-axis?
   1. A) $y = x^3$
   2. B) $y = x^2 + 1$
   3. C) $y = x + 1$
   4. D) $y = \frac{1}{x}$
2. If a graph is symmetric with respect to the y-axis and contains the point $(2, 5)$, which other point must also be on the graph?
   1. A) $(-2, -5)$
   2. B) $(2, -5)$
   3. C) $(-2, 5)$
   4. D) $(5, 2)$
3. Which function satisfies the condition $f(x) = f(-x)$?
   1. A) $f(x) = x^3 + x$
   2. B) $f(x) = x^2 + x$
   3. C) $f(x) = x^2 + |x|$
   4. D) $f(x) = x^3 + 1$
4. Which of the following graphs is NOT symmetric with respect to the y-axis?
   1. A) $x^2 + y^2 = 4$
   2. B) $y = |x|$
   3. C) $y = x^4$
   4. D) $y = x^3$
5. Determine if the graph of $y = cos(x)$ is symmetric with respect to the y-axis.
   1. A) Yes
   2. B) No
   3. C) Cannot be determined
   4. D) Only symmetric about the x-axis
6. Which equation's graph has y-axis symmetry?
   1. A) $y = (x-1)^2$
   2. B) $y = 2x$
   3. C) $y = x^2 + 3$
   4. D) $y = \frac{1}{x+1}$
7. The graph of $y=f(x)$ is symmetric with respect to the y-axis. If $f(3) = 7$, what is the value of $f(-3)$?
   1. A) -7
   2. B) 3
   3. C) 7
   4. D) -3