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Magazine Chapter 4: Problem 7
Sketch the graph of each function and spec. ify all \(x\) - and \(y\) -intercepts. $$y=-(x-1)^{4}$$
Short Answer
Expert verified
The x-intercept is at (1, 0) and the y-intercept is at (0, -1).
Step by step solution
01
Understand the Function
The given function is \(y = -(x-1)^4\), which is a transformed version of the parent function \(y = x^4\). This function involves a reflection across the x-axis due to the negative sign and is shifted 1 unit to the right because of \((x - 1)\).
02
Find the y-intercept
To find the y-intercept, set \(x = 0\):\[\begin{align*}y &= -(0-1)^4 \y &= -1^4 \y &= -1\end{align*}\]The y-intercept is at the point (0, -1).
03
Find the x-intercepts
To find the x-intercepts, set \(y = 0\) and solve for \(x\):\[-(x-1)^4 = 0\]This implies \[(x-1)^4 = 0 \]Taking the fourth root on both sides, we find \(x - 1 = 0\), so \(x = 1\). Thus, the x-intercept is at the point (1, 0).
04
Analyze the Graph
The function is a downward-opening parabola (quartic) which is symmetric around \(x = 1\). Since \((x-1)^4\) is always non-negative, the graph remains below the x-axis due to the negative sign. It dips at the vertex at \(x = 1\) and approaches zero as \(x\) moves away from 1.
05
Sketch the Graph
First plot the intercepts: (0, -1) for the y-intercept, and (1, 0) for the x-intercept. The vertex is at (1,0), the lowest point on the graph. The graph should curve downwards symmetrically on both sides of \(x = 1\), approaching the x-axis as \(x\) moves away from \(1\) in both directions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quartic Function
A quartic function is a polynomial function of degree four. It is expressed generally as: \[ f(x) = ax^4 + bx^3 + cx^2 + dx + e \]The highest exponent here is four, hence the name "quartic." Quartic functions can take on a variety of shapes. They may have up to four real roots or zero, depending on the specific equation.
In the given exercise, the function \( y = -(x-1)^4 \) is a form of quartic function. Here, the leading coefficient (the coefficient of \(x^4\)) is negative. This negative sign indicates that the graph of the quartic function will open downwards.
In the given exercise, the function \( y = -(x-1)^4 \) is a form of quartic function. Here, the leading coefficient (the coefficient of \(x^4\)) is negative. This negative sign indicates that the graph of the quartic function will open downwards.
- The equation suggests a transformation of the basic quartic function \( y = x^4 \).
- The graph is reflected over the x-axis and horizontally shifted one unit to the right.
X-intercept
The x-intercept of a function is the point where the graph intersects the x-axis. At this point, the value of \(y\) is zero. Finding the x-intercept involves setting the equation equal to zero and solving for \(x\).
For the function \( y = -(x-1)^4 \), the x-intercept is found by solving the equation:\[-(x-1)^4 = 0\]
Which simplifies to:\[(x-1)^4 = 0\]Upon taking the fourth root, we find:
\[ x - 1 = 0 \]In this case, the x-intercept is at \((1, 0)\). This indicates that the graph of the function touches the x-axis at exactly one point. It does not cross but rather touches and bounces back downwards because the function is reflected across the x-axis.
For the function \( y = -(x-1)^4 \), the x-intercept is found by solving the equation:\[-(x-1)^4 = 0\]
Which simplifies to:\[(x-1)^4 = 0\]Upon taking the fourth root, we find:
\[ x - 1 = 0 \]In this case, the x-intercept is at \((1, 0)\). This indicates that the graph of the function touches the x-axis at exactly one point. It does not cross but rather touches and bounces back downwards because the function is reflected across the x-axis.
Y-intercept
The y-intercept of a function is where the graph intersects the y-axis. At this point, the value of \(x\) is zero. Finding the y-intercept involves substituting \(x = 0\) into the equation and solving for \(y\).
For \( y = -(x-1)^4 \), finding the y-intercept involves calculating:\[y = -(0-1)^4 \]\[y = -1^4 \]This simplifies to:\[y = -1 \]Hence, the y-intercept is at the point \((0, -1)\). The graph crosses the y-axis at this point with the curve descending to \(-1\). It confirms that as \(x\) approaches zero, the function will start at \(-1\) and curve upward towards the x-intercept as it moves left to right.
For \( y = -(x-1)^4 \), finding the y-intercept involves calculating:\[y = -(0-1)^4 \]\[y = -1^4 \]This simplifies to:\[y = -1 \]Hence, the y-intercept is at the point \((0, -1)\). The graph crosses the y-axis at this point with the curve descending to \(-1\). It confirms that as \(x\) approaches zero, the function will start at \(-1\) and curve upward towards the x-intercept as it moves left to right.
Reflection Transformation
Reflection transformations involve flipping the graph of a function over a line, such as the x-axis or y-axis. This change alters the graph's appearance but not its overall shape.
In the case of the function \( y = -(x-1)^4 \), the negative sign in front of the quartic expression causes the reflection over the x-axis.
In the case of the function \( y = -(x-1)^4 \), the negative sign in front of the quartic expression causes the reflection over the x-axis.
- This means every point on the graph is mirrored across the x-axis.
- If the original graph were shaped like a U (upward opening), the reflection would cause it to point downwards like an inverted U.
- This transformation doesn鈥檛 change the intercept points, just the orientation.
