Vertical shift is a transformation of a function that involves moving the entire graph of the function up or down on the coordinate plane without changing the shape or orientation of the graph. This concept is applicable to a variety of function types, including linear, quadratic, exponential, logarithmic, and trigonometric functions.
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Vertical shift is often represented by the addition or subtraction of a constant value to the function, such as $f(x) = x^2 + 3$ or $g(x) = 2^x - 5$.
Vertical shift can affect the range and intercepts of a function, but it does not change the domain or the basic shape of the graph.
In the context of quadratic functions, a vertical shift changes the y-intercept of the parabola without affecting the vertex or the axis of symmetry.
For exponential and logarithmic functions, a vertical shift can change the y-intercept and the rate of growth or decay, but it does not alter the base of the function.
Vertical shift is an important concept in modeling real-world phenomena, as it allows for the adjustment of the output values of a function to better fit the data.
Review Questions
Explain how a vertical shift affects the graph of a quadratic function.
When a quadratic function undergoes a vertical shift, the parabolic shape of the graph remains the same, but the entire graph is moved up or down on the coordinate plane. This changes the y-intercept of the function, but it does not affect the location of the vertex or the axis of symmetry. For example, if the function $f(x) = x^2$ is vertically shifted up by 3 units, the new function would be $g(x) = x^2 + 3$, where the graph is moved up 3 units without changing the shape or orientation of the parabola.
Describe the impact of a vertical shift on the graph of an exponential function.
A vertical shift of an exponential function $f(x) = a^x$ results in a new function $g(x) = a^x + b$, where $b$ is the vertical shift. This change affects the y-intercept of the graph, as the function is shifted up or down on the coordinate plane. However, the base of the exponential function, $a$, remains the same, so the rate of growth or decay is not altered. The overall shape of the exponential curve is preserved, but its position on the y-axis is changed by the vertical shift.
Analyze how a vertical shift influences the graph of a trigonometric function, such as the sine or cosine function.
When a trigonometric function, like $f(x) = ext{sin}(x)$ or $g(x) = ext{cos}(x)$, undergoes a vertical shift, the amplitude and period of the function remain the same, but the entire graph is moved up or down on the coordinate plane. This vertical shift is represented by the addition or subtraction of a constant value to the function, such as $h(x) = ext{sin}(x) + 3$ or $k(x) = ext{cos}(x) - 2$. The vertical shift changes the y-intercept and the range of the trigonometric function, but it does not affect the frequency or the horizontal positioning of the graph.