11th Grade Q1

To find the $y$-intercept of a function involving a logarithm, you determine the point where the graph crosses the $y$-axis. This occurs when the input value is $x=0$.

An example of such a function is:

where:

• $a$ is a constant multiplier,
• $b$ is the base of the logarithm,
• $x+d$ is the argument of the logarithm,
• $c$ is a constant shift.

The $y$-intercept exists if $x=0$ is within the domain of the function.

Method

1. Check the domain

The logarithmic function is defined only when the argument inside the logarithm is positive. For $f(x)=a\cdot {\log }_b(x+d)+c$, this means you need $d>0$.

2. Substitute $x=0$

If $x=0$ is within the domain, substitute it into the function:

Simplify to:

3. Write the intercept as a point

The $y$-intercept is written as:

Example

Find the $y$-intercept of $f(x)=3\cdot {\log }_2(x+4)-2$.

1. Check the domain:
   The function is defined for $x+4>0$, or $x>-4$. At $x=0$, $x+4=4$, so the function is defined.
2. Substitute $x=0$:
   Since ${\log }_2(4)=2$:
   

1. Write the $y$-intercept:
   The $y$-intercept is:
   

Key Points to Remember

• A logarithmic function has a $y$-intercept only if $x=0$ is within the domain of the function.
• The $y$-intercept is always written as a point $(0,f(0))$.