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Magazine Chapter 4: Problem 6
Sketch the graph of each function and spec. ify all \(x\) - and \(y\) -intercepts. $$y=-3 x^{4}$$
Short Answer
Expert verified
The graph has x- and y-intercepts at (0,0) and has ends pointing downwards.
Step by step solution
01
Understand the function type
The given function is a polynomial function in terms of \(x\). Specifically, it is a quartic function (degree 4) with a negative leading coefficient.
02
Determine the y-intercept
To find the \(y\)-intercept, set \(x = 0\) in the function. Substitute into the equation: \(y = -3(0)^4 = 0\). The \(y\)-intercept is at (0,0).
03
Determine the x-intercepts
For the \(x\)-intercepts, set \(y = 0\). The equation becomes \(-3x^4 = 0\), which simplifies to \(x^4 = 0\). The only solution is \(x = 0\). Thus, the \(x\)-intercept is also at (0,0).
04
Analyze the end behavior
Since this is a quartic polynomial with a negative leading coefficient, the ends of the graph will both point downwards. As \(x\to \pm\infty\), \(y\to -\infty\).
05
Sketch the graph
Plot the intercept (0,0), and note that as \(x\) moves away from 0 in either direction, \(y\) becomes increasingly negative. The graph is symmetrical around the \(y\)-axis (even function), and it returns to negative infinity as \(x\) moves towards positive or negative infinity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quartic Functions
Quartic functions are polynomial functions with a degree of 4. This means the highest power of the variable, typically denoted as \( x \), is raised to the fourth power. These functions can produce a wide variety of graph shapes but generally showcase a single dominant shape pattern. Within quartic functions, the key features include:
- Coefficient Sign: The sign of the leading coefficient dictates the direction of the graph's ends. A positive leading term makes the ends of the graph open upwards, while a negative term, like in our example of \( y = -3x^4 \), causes the graph to face downwards.
- Graph Symmetry: Quartic functions can be symmetric, often even suggesting that the graph might resemble a bowl (opened up or down) or an "M" or "W" shape depending on other terms.
- Turning Points: Typically, these functions have multiple turning points, depending on specific terms, which makes them visually interesting in graph representation.
x-intercept
To determine the x-intercept of a polynomial function, you set \( y = 0 \) and solve for \( x \). The x-intercept is the point where the graph crosses the x-axis, meaning the output value at this point is zero. For the function \( y = -3x^4 \):
- The equation is set to zero: \(-3x^4 = 0\).
- Solving for \( x \) gives \( x^4 = 0 \), which clearly indicates that \( x = 0 \).
- Therefore, there's only one x-intercept, at the point (0,0).
y-intercept
Unlike the x-intercept, to determine the y-intercept, you set \( x = 0 \) in the function. For the function \( y = -3x^4 \):
- You plug in \( x = 0 \): \( y = -3(0)^4 \).
- The calculation simplifies to \( y = 0 \), making the y-intercept point (0,0).
End Behavior
End behavior describes how the graph of a polynomial behaves as \( x \) approaches positive or negative infinity. This is particularly useful for understanding how a graph behaves outside its local neighborhood:
- Negative Leading Coefficient: In our example, the negative leading coefficient \(-3\) indicates that the ends of the graph will both point downwards.
- Behavior Near Infinity: As \( x \to \pm\infty \), \( y \to -\infty \). This tells us that both ends of the graph drop towards negative infinity.
- Symmetry and Consistent Behavior: As quartic functions with negative coefficients exhibit this consistent end behavior, sketching becomes intuitive. The graph starts and ends low, with the possible rise in middle sections depending on other terms.
