To determine which equation has the least steep graph, we need to compare the slopes of the given equations. The slope of a line in the equation form [tex]\( y = mx + b \)[/tex] is represented by [tex]\( m \)[/tex].
Here are the steps to find the solution:
1. Identify the slopes from each equation:
- For equation [tex]\( A \)[/tex]: [tex]\( y = 2x - 7 \)[/tex]
- The slope [tex]\( m \)[/tex] is 2.
- For equation [tex]\( B \)[/tex]: [tex]\( y = \frac{1}{4} x + 9 \)[/tex]
- The slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
- For equation [tex]\( C \)[/tex]: [tex]\( y = -\frac{1}{2} x - 3 \)[/tex]
- The slope [tex]\( m \)[/tex] is [tex]\( -\frac{1}{2} \)[/tex].
- For equation [tex]\( D \)[/tex]: [tex]\( y = -6 x + 1 \)[/tex]
- The slope [tex]\( m \)[/tex] is -6.
2. Compare the absolute values of the slopes to determine steepness:
- The absolute value of the slope for equation [tex]\( A \)[/tex] is [tex]\( |2| = 2 \)[/tex].
- The absolute value of the slope for equation [tex]\( B \)[/tex] is [tex]\( \left|\frac{1}{4}\right| = \frac{1}{4} \)[/tex].
- The absolute value of the slope for equation [tex]\( C \)[/tex] is [tex]\( \left| -\frac{1}{2} \right| = \frac{1}{2} \)[/tex].
- The absolute value of the slope for equation [tex]\( D \)[/tex] is [tex]\( |-6| = 6 \)[/tex].
3. Identify the smallest absolute value of the slope:
- Compare 2, [tex]\( \frac{1}{4} \)[/tex], [tex]\( \frac{1}{2} \)[/tex], and 6.
- The smallest absolute value is [tex]\( \frac{1}{4} \)[/tex].
4. Determine which equation corresponds to this smallest absolute value:
- The slope [tex]\( \frac{1}{4} \)[/tex] belongs to equation [tex]\( B \)[/tex].
Therefore, the equation with the least steep graph is:
[tex]\[ \boxed{B} \][/tex]
