91Ó°ÊÓ

Understanding the slope-intercept form is crucial for graphing linear equations. The formula is written as: \ \( y = mx + b \). \ Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept. \ In our exercise, the equation \( 5y = 4x + 5 \) is given. To convert it, we divide both sides by 5: \ \( y = \frac{4}{5}x + 1 \). \ Now it's in slope-intercept form, where \( \frac{4}{5} \) is the slope and \( 1 \) is the y-intercept. \ This tells us that for every 5 units moved horizontally, the line moves 4 units vertically.

Symmetry in graphs helps us understand the balance and shape of equations. We check for three types of symmetry: \

• \( x \)-axis symmetry: Each point \((x, y)\) should have a corresponding point \((x, -y)\).
• \( y \)-axis symmetry: Each point \((x, y)\) should have a corresponding point \((-x, y)\).
• Origin symmetry: Each point \((x, y)\) should have a corresponding point \((-x, -y)\).

\ To visually check for symmetry, we graph the line. For our line, it becomes clear it does not mirror across any axis or the origin. \ Later, we will verify these observations algebraically.

Graphing an equation on the coordinate plane helps visualize the relationships between variables. Let's analyze our equation. \ After converting to slope-intercept form \(y = \frac{4}{5}x + 1\), we start plotting from the y-intercept \((0,1)\). Using the slope \( \frac{4}{5} \), we plot another point by moving 4 units up and 5 units right. \ Connecting these points creates a line that slopes upward, showing that for each increase in x, y also increases. \ This geometric representation assists in understanding the nature of the function.

After visual inspection, we use algebraic methods to verify symmetry. \ For \( x \)-axis symmetry, replace \( y \) with \( -y \) in the original equation and check if it matches: \ \[ 5(-y) = 4x + 5 \] \[ -5y = 4x + 5 \]. \ The result shows an inconsistency, meaning no symmetry with \( x \)-axis. \ Next, we check for \( y \)-axis symmetry by replacing \( x \) with \( -x \): \ \[ 5y = 4(-x) + 5 \] \[ 5y = -4x + 5 \]. \ Here too, the equations don't match, ruling out \( y \)-axis symmetry. \ Finally, for origin symmetry, replace both \( x \) and \( y \) with their negatives: \ \[ 5(-y) = 4(-x) + 5 \] \[ -5y = -4x + 5 \] \[ 5y = 4x - 5 \]. \ Again, the result is different from the original, confirming no origin symmetry. This thorough checking solidifies our understanding.