The graph of the derivative f ' of a function f is shown. WebAssign Plot (a) On what interval is f increasing? (Enter your answer using interval notation.) ? On what intervals is f decreasing? (Enter your answer using interval notation.) ? (b) At what values of x does f have a local maximum or minimum? (Enter your answers as a comma-separated list.) x = ?
The graph of the derivative f ' of a function f is shown. WebAssign Plot (a) On what interval is f increasing? (Enter your answer using interval notation.) ? On what intervals is f decreasing? (Enter your answer using interval notation.) ? (b) At what values of x does f have a local maximum or minimum? (Enter your answers as a comma-separated list.) x = ?
The graph of the derivative f ' of a function f is shown. WebAssign Plot (a) On what interval is f increasing? (Enter your answer using interval notation.) ? On what intervals is f decreasing? (Enter your answer using interval notation.) ? (b) At what values of x does f have a local maximum or minimum? (Enter your answers as a comma-separated list.) x = ?
The graph of the derivative f ' of a function f is shown.
WebAssign Plot
(a) On what interval is f increasing? (Enter your answer using interval notation.)
?
On what intervals is f decreasing? (Enter your answer using interval notation.)
?
(b) At what values of x does f have a local maximum or minimum? (Enter your answers as a comma-separated list.)
x = ?
Transcribed Image Text:### Understanding Parabolas in Quadratic Functions
The image above illustrates a graph of a parabola, which is a fundamental concept in the study of quadratic functions. This particular graph is a visual representation of a quadratic equation in the standard form \( y = ax^2 + bx + c \).
**Key Features of the Graph:**
1. **Axis Labels:**
- The horizontal axis is labeled as the "x" axis.
- The vertical axis is not labeled with specific values in this image, but it typically represents the "y" values in the coordinate plane.
2. **Shape of the Parabola:**
- The parabola opens downwards, which indicates that the quadratic equation has a leading coefficient (the "a" value in \( y = ax^2 + bx + c \)) that is negative.
- The vertex (the highest point on the graph) is at approximately \((8, 6)\). This point represents the maximum value of the quadratic function.
3. **X-Intercepts:**
- The points where the parabola intersects the x-axis are known as the x-intercepts or roots. In this graph, the x-intercepts appear to be at \( x = 2 \) and \( x = 14 \). These are the solutions to the quadratic equation when \( y = 0 \).
4. **Symmetry:**
- The parabola is symmetric with respect to a vertical line that passes through the vertex. This line is called the axis of symmetry and for this graph is at \( x = 8 \).
Understanding these components is essential for analyzing and interpreting the behavior of quadratic functions. Parabolas are used in various real-world applications such as physics, engineering, and economics, making their study not only academically significant but also practically valuable.
For further exploration, students can practice plotting quadratic functions with different coefficients, observe how these changes affect the shape and position of the parabola, and deepen their understanding of vertex form and factored form of quadratic equations.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
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