Question

What will happen to the graph of the function \(f(x)=4 x^{2}-18\) if it is transformed into the function \(g(x)=4(x-2)^{2}-15 ?\) (A) It will shift down 2 units and shift to the left 3 units. (B) It will shift up 3 units and shift to the right 2 units. (C) It will shift up 2 units and shift to the left 3 units. (D) It will shift down 3 units and shift to the right 2 units.

Short answer

(B) The function shifts 3 units up and 2 units to the right.

Step-by-step solution

Identify the Original Function

The original function given is \( f(x) = 4x^2 - 18 \). This is a quadratic function in standard form, which can also be written in vertex form as \( f(x) = 4(x - 0)^2 - 18 \). The vertex is at the point \((0, -18)\).

Identify the Transformed Function

The transformed function is \( g(x) = 4(x-2)^2 - 15 \), which is already in vertex form. This means the vertex has shifted to the point \((2, -15)\).

Determine Horizontal Shift

Compare the \(x\)-coordinates of the vertices of \(f(x)\) and \(g(x)\). The transformation \(4(x-2)^2\) implies a horizontal shift to the right by 2 units, because \(x\) is replaced by \((x-2)\).

Determine Vertical Shift

Compare the constant terms in \(f(x)\) and \(g(x)\). The move from \(-18\) to \(-15\) suggests a vertical shift upwards by 3 units, since \(-15\) is more than \(-18\).

Identify the Overall Transformation

The function \(g(x)\) represents a graph that has shifted 2 units to the right and 3 units up compared to the graph of \(f(x)\).

Key concepts

Vertex Form

The vertex form of a quadratic function is an essential tool when working with transformations. It is written as \(y = a(x-h)^2 + k\), where

• \(a\) determines the direction and width of the parabola.
• \(h\) represents the horizontal shift.
• \(k\) is the vertical shift.

This form clearly shows the vertex of the parabola, located at \((h, k)\). By examining the vertex form, students can easily identify how a graph moves from its original position. For example, in the function \(f(x) = 4(x - 0)^2 - 18\), the vertex is at (0, -18). In the transformed function \(g(x) = 4(x-2)^2 - 15\), it is at (2, -15). The vertex form makes tracking these changes straightforward.

Graph Shifting

Graph shifting is a transformation that moves a function's graph in a plane without altering its shape. It is crucial for understanding how the function itself remains unchanged while its position modifies. This involves two main types of shifts: horizontal and vertical. Horizontal shifts adjust the \(x\)-coordinate while vertical shifts affect the \(y\)-coordinate. By examining the changes in these coordinates in the vertex form, students can see how a graph is shifted on the plane. Both types of shifts are evident in our example. Here, the movement from \((0, -18)\) to \((2, -15)\) shows that the graph moves 2 units to the right and 3 units up. Recognizing these is instrumental in mastering graph transformations.

Horizontal Shift

A horizontal shift occurs when a function's graph slides left or right on the Cartesian plane. In the vertex form \(y = a(x-h)^2 + k\), the horizontal shift is determined by the value of \(h\).

• If \(h > 0\), the graph shifts to the right.
• If \(h < 0\), it shifts to the left.

In the transformation from \(f(x) = 4(x - 0)^2 - 18\) to \(g(x) = 4(x-2)^2 - 15\), the replacement of \(x\) with \((x-2)\) indicates a horizontal shift 2 units to the right. Understanding how to read this part of the function in vertex form is key to predicting and explaining horizontal movements on a graph.

Vertical Shift

Vertical shift involves moving a graph up or down along the \(y\)-axis. In the vertex form \(y = a(x-h)^2 + k\), the vertical shift is controlled by the value \(k\).

• If \(k > 0\), the graph will shift upwards.
• If \(k < 0\), it moves downwards.

In our transformation from the function \(f(x) = 4x^2 - 18\) to \(g(x) = 4(x-2)^2 - 15\), comparing the constant terms \(-18\) and \(-15\) reveals the vertical shift upwards by 3 units. Recognizing the change in the constant term in vertex form aids students in determining how the graph will shift vertically, providing a comprehensive view of the graph's new position.