Question

perform the given transformation on point d and then on its image.
original point
d
(-5, -4)
image
d
(blank)

Explanation:

Step1: Identify the transformation (assuming reflection over y-axis, common transformation; if not, need more info, but let's proceed with reflection over y-axis as a common case). The rule for reflection over y-axis is \((x,y) \to (-x,y)\).

Original point \(D = (-5, -4)\). Applying reflection over y-axis: \(x\)-coordinate becomes \(-(-5)=5\), \(y\)-coordinate remains \(-4\). So image \(D'=(5, -4)\).
(Note: If the transformation was different, like translation, rotation, etc., but since the problem says "perform the given transformation" and no specific is stated here, but from the grid, maybe reflection or translation. Wait, maybe the intended transformation is reflection over y-axis or x-axis? Wait, let's check the grid. The original point is at (-5, -4) (wait, in the grid, the x-axis and y-axis: wait, the grid has x-axis vertical? Wait, no, standard coordinate system: x horizontal, y vertical. Wait, the grid labels: the horizontal axis is y? Wait, no, the axes are labeled: the horizontal arrow is y (from -10 to 10), vertical arrow is x (from -10 to 10). So the coordinates: x is vertical, y is horizontal. So point D: x-coordinate (vertical) is -5? Wait, no, in standard math, x is horizontal, y vertical. But here, the vertical axis is labeled x (with arrow down/up), horizontal axis labeled y (arrow left/right). So point D: looking at the grid, the horizontal (y) coordinate: from origin (0,0), moving left 5 units (y=-5? Wait, no, the horizontal axis (y) has 0 at center, right is positive y, left negative y. Vertical axis (x) has 0 at center, up positive x, down negative x. So point D: horizontal (y) is -5? Wait, no, the dot is at horizontal position (y) of -5? Wait, no, the grid: each square is 1 unit. The dot is at x (vertical) of -5? Wait, the original point is given as (-5, -4). So x=-5 (vertical), y=-4 (horizontal). So if we reflect over the y-axis (horizontal axis), the y-coordinate (horizontal) changes sign. Wait, maybe the coordinate system here has x vertical and y horizontal. So reflection over y-axis (horizontal axis) would change the y-coordinate's sign. So original \(D = (-5, -4)\) (x=-5, y=-4). Reflect over y-axis: y becomes -(-4)=4? No, that doesn't make sense. Wait, maybe the problem has a typo, or the intended transformation is reflection over x-axis. Rule for reflection over x-axis: \((x,y) \to (x, -y)\). So original \(D=(-5, -4)\), reflection over x-axis: \(y\)-coordinate becomes -(-4)=4, so \(D'=(-5, 4)\). But without the specific transformation, it's ambiguous. Wait, maybe the problem was to reflect over y-axis. Let's re-express the coordinate system: in standard terms, if the horizontal axis is x and vertical is y, but here the labels are swapped. So maybe the horizontal axis is x, vertical is y. Wait, the image shows the horizontal arrow labeled y (from -10 to 10), vertical arrow labeled x (from -10 to 10). So x is vertical, y is horizontal. So point D: x (vertical) is -5 (down 5 units from origin), y (horizontal) is -4 (left 4 units from origin)? No, the dot is at x (vertical) of -5? Wait, the original point is given as (-5, -4). So x=-5, y=-4. Let's assume the transformation is reflection over the y-axis (horizontal axis, y-axis). So reflection over y-axis: (x, y) → (x, -y). Wait, no, reflection over y-axis (horizontal line) would be (x, y) → (x, -y) (reflection over x-axis is (x,y)→(x,-y), reflection over y-axis (vertical line) is (x,y)→(-x,y)). Wait, I think the axes are labeled incorrectly in the image, or it's a non-standard coordinate system. Assuming standard coordinate system (x horizontal, y vertical), then original point D is (-5, -4) (x=-5, y=-4). If we perform a reflection over the y-axis (vertical line x=0), the rule is (x,y)→(-x,y). So x becomes -(-5)=5, y remains -4. So D'=(5, -4).

Step2: Confirm the transformation. Since the problem says "perform the given transformation" but the transformation isn't stated, but from typical problems, reflection over y-axis is common. So applying the rule, we get the image.

Answer:

If reflection over y-axis, the image \(D'\) is \((5, -4)\). (If the transformation was different, adjust accordingly, but with the given info, this is a plausible solution.)