91Ó°ÊÓ

The vertical intercept, also known as the y-intercept, is where the graph of a function crosses the y-axis. To find this intercept, you set the input variable \( x \) to zero and calculate the output \( f(x) \). Imagine this like asking "Where, along the vertical line where \( x = 0 \), does the graph hit?"

In the given function \( f(x) = -3|x - 2| - 1 \), we set \( x = 0 \):

• First, substitute \( x = 0 \) into the function: \( f(0) = -3|0 - 2| - 1 \).
• Simplify the absolute value expression \( |0 - 2| \) which equals 2.
• Now the expression becomes \( -3 \times 2 - 1 = -6 - 1 \).
• This gives \( f(0) = -7 \).

So, the vertical intercept of this function is the point \( (0, -7) \). This tells us that the graph crosses the y-axis at -7 on the y-axis.

Horizontal intercepts, or x-intercepts, are points where the graph of a function crosses the x-axis. This is where the output \( f(x) \) is zero. To find these intercepts, we set \( f(x) = 0 \) and solve for \( x \).

In the context of our function \( f(x) = -3|x - 2| - 1 \), finding the horizontal intercept involves:

• Setting the equation to zero: \( 0 = -3|x - 2| - 1 \).
• Add 1 to both sides to isolate the absolute value term: \( 1 = -3|x - 2| \).
• Next, divide both sides by -3: \( |x - 2| = -\frac{1}{3} \).

Here's the catch: the absolute value of a number, noted as \( |x| \), is always non-negative. So, \( |x - 2| = -\frac{1}{3} \) has no real solutions because a negative result is impossible here. Therefore, this function has no horizontal intercepts.

An absolute value function involves the absolute value operator \( |x| \), which makes any number non-negative. For example, both \( |3| \) and \( |-3| \) equal 3. The absolute value measures the distance of a number from zero on the number line, ignoring direction.

In our example, the absolute value function is expressed within \( f(x) = -3|x - 2| - 1 \). The term \( |x - 2| \) indicates that each input \( x \) is adjusted by subtracting 2, centering the distortion around \( x = 2 \). Here's how the function impacts graph behavior:

• It translates the basic graph of \( |x| \) horizontally by 2 units to the right.
• The multiplier of -3 flips the graph upside down (due to the negative sign) and stretches it, making it wider.
• The \(-1\) at the end moves the graph down by 1 unit.

Absolute value functions create a distinctive "V" shape on the graph. The transformations modify this shape to match the specific function being analyzed.