Question

click on the graph to plot a point. click a point to delete it.

Explanation:

To solve the problem of reflecting a point over the \( x \)-axis, we follow these steps:

Step 1: Recall the reflection rule over the \( x \)-axis

The rule for reflecting a point \( (x, y) \) over the \( x \)-axis is \( (x, -y) \). This means we keep the \( x \)-coordinate the same and change the sign of the \( y \)-coordinate.

Step 2: Identify the original point

From the graph, we can see that the original point \( F \) is at \( (-6, 1) \) (assuming the coordinates from the graph). Wait, actually, looking at the graph, the point \( F \) is at \( (-6, 1) \)? Wait, no, let's check the axes. The \( x \)-axis is vertical and \( y \)-axis is horizontal? Wait, no, standard coordinate system: \( x \)-axis is horizontal (left-right), \( y \)-axis is vertical (up-down). Wait, in the given graph, the \( x \)-axis is vertical (labeled \( x \) with arrow down at bottom) and \( y \)-axis is horizontal (labeled \( y \) with arrow right at right). So it's a rotated coordinate system? Wait, no, maybe it's a typo or mislabeling. Wait, the standard is \( x \)-horizontal, \( y \)-vertical. But in the given graph, the vertical axis is labeled \( x \) (with arrow down) and horizontal axis labeled \( y \) (with arrow right). So maybe it's a reflection where the \( x \)-axis (vertical) is the axis of reflection. Wait, the problem says "reflected over the \( x \)-axis". Let's assume the standard: if the point is \( (x, y) \) with \( x \)-horizontal, \( y \)-vertical, then reflection over \( x \)-axis is \( (x, -y) \). But in the graph, the vertical axis is \( x \), horizontal is \( y \). So maybe the point \( F \) is at \( (1, -6) \)? Wait, no, let's look at the grid. The vertical axis (labeled \( x \)) has ticks at -6, -5, ..., 0, 1, ..., 6 (arrow down at bottom, so positive \( x \) is down). The horizontal axis (labeled \( y \)) has ticks at -6, -5, ..., 0, 1, ..., 6 (arrow right at right, so positive \( y \) is right). The point \( F \) is at \( (x, y) = (1, -6) \)? Wait, no, the point \( F \) is at the left side, at \( y = -6 \) (horizontal axis) and \( x = 1 \) (vertical axis)? Wait, this is confusing. Wait, maybe the graph is labeled with \( x \)-axis vertical (up-down) and \( y \)-axis horizontal (left-right), which is non-standard, but let's proceed.

Assuming the original point \( F \) is at \( (x, y) = (1, -6) \) (since it's at \( x = 1 \) (vertical) and \( y = -6 \) (horizontal)). Wait, no, the user says "reflected over the \( x \)-axis". Let's use the standard reflection rule: if the point is \( (a, b) \), reflection over \( x \)-axis is \( (a, -b) \).

Wait, maybe the original point is \( (-6, 1) \) in standard coordinates ( \( x \)-horizontal, \( y \)-vertical). Let's assume that. So reflecting \( (-6, 1) \) over \( x \)-axis:

Step 3: Apply the reflection rule

For a point \( (x, y) \), reflection over \( x \)-axis is \( (x, -y) \). So if original point is \( (-6, 1) \), then reflected point is \( (-6, -1) \).

But wait, the graph shows the point \( F \) at \( (-6, 1) \) (assuming \( x \)-horizontal, \( y \)-vertical). So to reflect over \( x \)-axis, we change the \( y \)-coordinate sign.

So:

• Original point: \( (-6, 1) \)
• Reflected point: \( (-6, -1) \)

Now, we need to plot this point on the graph. The \( x \)-coordinate is -6 (left 6 units on horizontal axis), \( y \)-coordinate is -1 (down 1 unit on vertical axis).

Final Answer

The reflected point over the \( x \)-axis is \( \boldsymbol{(-6, -1)} \).

Answer:

To solve the problem of reflecting a point over the \( x \)-axis, we follow these steps:

Step 1: Recall the reflection rule over the \( x \)-axis

The rule for reflecting a point \( (x, y) \) over the \( x \)-axis is \( (x, -y) \). This means we keep the \( x \)-coordinate the same and change the sign of the \( y \)-coordinate.

Step 2: Identify the original point

From the graph, we can see that the original point \( F \) is at \( (-6, 1) \) (assuming the coordinates from the graph). Wait, actually, looking at the graph, the point \( F \) is at \( (-6, 1) \)? Wait, no, let's check the axes. The \( x \)-axis is vertical and \( y \)-axis is horizontal? Wait, no, standard coordinate system: \( x \)-axis is horizontal (left-right), \( y \)-axis is vertical (up-down). Wait, in the given graph, the \( x \)-axis is vertical (labeled \( x \) with arrow down at bottom) and \( y \)-axis is horizontal (labeled \( y \) with arrow right at right). So it's a rotated coordinate system? Wait, no, maybe it's a typo or mislabeling. Wait, the standard is \( x \)-horizontal, \( y \)-vertical. But in the given graph, the vertical axis is labeled \( x \) (with arrow down) and horizontal axis labeled \( y \) (with arrow right). So maybe it's a reflection where the \( x \)-axis (vertical) is the axis of reflection. Wait, the problem says "reflected over the \( x \)-axis". Let's assume the standard: if the point is \( (x, y) \) with \( x \)-horizontal, \( y \)-vertical, then reflection over \( x \)-axis is \( (x, -y) \). But in the graph, the vertical axis is \( x \), horizontal is \( y \). So maybe the point \( F \) is at \( (1, -6) \)? Wait, no, let's look at the grid. The vertical axis (labeled \( x \)) has ticks at -6, -5, ..., 0, 1, ..., 6 (arrow down at bottom, so positive \( x \) is down). The horizontal axis (labeled \( y \)) has ticks at -6, -5, ..., 0, 1, ..., 6 (arrow right at right, so positive \( y \) is right). The point \( F \) is at \( (x, y) = (1, -6) \)? Wait, no, the point \( F \) is at the left side, at \( y = -6 \) (horizontal axis) and \( x = 1 \) (vertical axis)? Wait, this is confusing. Wait, maybe the graph is labeled with \( x \)-axis vertical (up-down) and \( y \)-axis horizontal (left-right), which is non-standard, but let's proceed.

Assuming the original point \( F \) is at \( (x, y) = (1, -6) \) (since it's at \( x = 1 \) (vertical) and \( y = -6 \) (horizontal)). Wait, no, the user says "reflected over the \( x \)-axis". Let's use the standard reflection rule: if the point is \( (a, b) \), reflection over \( x \)-axis is \( (a, -b) \).

Wait, maybe the original point is \( (-6, 1) \) in standard coordinates ( \( x \)-horizontal, \( y \)-vertical). Let's assume that. So reflecting \( (-6, 1) \) over \( x \)-axis:

Step 3: Apply the reflection rule

For a point \( (x, y) \), reflection over \( x \)-axis is \( (x, -y) \). So if original point is \( (-6, 1) \), then reflected point is \( (-6, -1) \).

But wait, the graph shows the point \( F \) at \( (-6, 1) \) (assuming \( x \)-horizontal, \( y \)-vertical). So to reflect over \( x \)-axis, we change the \( y \)-coordinate sign.

So:

• Original point: \( (-6, 1) \)
• Reflected point: \( (-6, -1) \)

Now, we need to plot this point on the graph. The \( x \)-coordinate is -6 (left 6 units on horizontal axis), \( y \)-coordinate is -1 (down 1 unit on vertical axis).

Final Answer

The reflected point over the \( x \)-axis is \( \boldsymbol{(-6, -1)} \).