Question

If the \(x\) -intercept of the graph of \(y=f(x)\) is located at \((a, 0)\) and the \(y\) -intercept is located at \((0, b),\) determine the \(x\) -intercept and \(y\) -intercept after the following transformations of the graph of \(y=f(x)\). a) \(y=-f(-x)\) b) \(y=2 f\left(\frac{1}{2} x\right)\) c) \(y+3=f(x-4)\) d) \(y+3=\frac{1}{2} f\left(\frac{1}{4}(x-4)\right)\)

Short answer

a) x-intercept: (-a, 0), y-intercept: (0, -b); b) x-intercept: (a, 0), y-intercept: (0, 2b); c) x-intercept: (a+4, -3), y-intercept: (0, f(-4)-3); d) x-intercept: (4a+4, -3), y-intercept: (0, \(\frac{1}{2}f(-1) - 3\)).

Step-by-step solution

Identify Intercepts of Original Function

The original function is given as \(y=f(x)\). The \(x\)-intercept is at \((a, 0)\) and the \(y\)-intercept is at \((0, b)\).

Transformation a: \(y=-f(-x)\)

For the transformation \(y=-f(-x)\), the graph reflects over the y-axis and then over the x-axis. The new \(x\)-intercept is found by setting \(y=0\): \( 0=-f(-x) \Rightarrow f(-x)=0 \Rightarrow x=-a \). The new \(y\)-intercept is \((0, -b)\). Therefore, the new intercepts are \((-a, 0)\) and \((0, -b)\).

Transformation b: \(y=2 f\left( \frac{1}{2} x \right)\)

For \(y=2 f\left( \frac{1}{2} x \right)\), the graph is horizontally stretched by a factor of 2 and vertically stretched by a factor of 2. The new \(x\)-intercept remains the same, \((a, 0)\), because it depends on where \(f(x)\) is zero. The \(y\)-intercept is scaled vertically by a factor of 2, so it is \((0, 2b)\).

Transformation c: \(y+3=f(x-4)\)

For \(y+3=f(x-4)\), the graph is shifted right by 4 units and down by 3 units. The \(x\)-intercept is previously \(f(x) = 0 \Rightarrow x=a\). So, with the transformation, the \(x\)-intercept is \((a+4, -3)\). The \(y\)-intercept is transformed by setting \(x=0\): \( y + 3 = f(-4) \Rightarrow y = f(-4) - 3\), giving a new \(y\)-intercept at \((0, f(-4)-3)\).

Transformation d: \(y+3=\frac{1}{2} f\left( \frac{1}{4}(x-4) \right)\)

For \(y+3=\frac{1}{2} f\left( \frac{1}{4}(x-4) \right)\), the graph is horizontally stretched by a factor of 4, shifted right by 4 units, vertically compressed by a factor of 2, and shifted down by 3 units. The \(x\)-intercept comes from \(f(x)=0\), so transforming \(x=a\) gives \(x=4a+4\) with \(y=-3\). Therefore, the \(x\)-intercept is \((4a+4, -3)\). The \(y\)-intercept is found by setting \(x=0\): \(y + 3 = \frac{1}{2} f(-1) \Rightarrow y = \frac{1}{2} f(-1) - 3\), which gives a new \(y\)-intercept at \((0, \frac{1}{2} f(-1) - 3)\).

Key concepts

x-intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. Mathematically, this happens when the value of y is 0. For a function y = f(x), if f(a) = 0, then the x-intercept is at (a, 0). Understanding how transformations affect the x-intercept is crucial. For example, if the transformation is y = -f(-x), the graph reflects over both the y-axis and the x-axis. This changes the x-intercept from (a, 0) to (-a, 0). If multiple transformations are combined, like shifts or stretches, the new x-intercept must be found considering each step.

y-intercept

The y-intercept of a graph is where the graph crosses the y-axis, which occurs when x is 0. For y = f(x), if f(0) = b, the y-intercept is at (0, b). Transformations can change this intercept. For instance, a transformation y = 2 f(x) vertically stretches the graph by a factor of 2. If the original y-intercept is (0, b), the new y-intercept becomes (0, 2b). Another example is y + 3 = f(x - 4), which shifts the graph down by 3 units and right by 4 units. In this case, you have to adjust the y-intercept by these shifts to find the new intercept coordinates.

graph transformations

Graph transformations modify a function's graph in different ways, and it's often a combination of transformations that changes the graph. Here are some common types:
- **Translation**: Shifting the graph horizontally or vertically.
- **Stretching/Compressing**: Making the graph narrower or wider (horizontally or vertically).
- **Reflection**: Flipping the graph across an axis. For example, with the transformation y = f(x) becoming y = -f(x), the graph flips over the x-axis. Each transformation has predictable effects on the intercepts and the overall shape of the graph.

horizontal stretch

A horizontal stretch changes the width of the graph by a specific factor. For y = f(ax), if 0 < a < 1, the graph expands horizontally. If a > 1, it compresses. Take y = f(1/2x), which stretches the graph by a factor of 2 horizontally. The x-intercept usually remains unchanged, but the y-values shift according to the stretch factor. It's important to adjust both intercepts and other key points to understand the new shape of the graph.

vertical stretch

A vertical stretch alters a graph's height. In the transformation y = kf(x), if k > 1, the graph stretches vertically. If 0 < k < 1, it compresses. For instance, in y = 2f(x), the graph stretches vertically by a factor of 2. This changes the y-intercept from (0, b) to (0, 2b), but the x-intercept remains at (a, 0). Understanding vertical stretches helps predict how the graph's heights change while keeping the x-intercept positions steady.

reflection

Reflection flips the graph over an axis. There are two common types:
- **Over the x-axis**: y = -f(x) turns each y-value into its opposite.
For example, if y = 3 when x = 1, then y = -3 after reflection.
- **Over the y-axis**: y = f(-x) flips the graph over the y-axis.
Example: The point (1, 3) would move to (-1, 3). Combining these reflections can invert the graph both vertically and horizontally, affecting the intercepts<|image_sentinel|>. For instance, y = -f(-x) reflects the graph over both axes.