Write the equation of each function shown below in vertex and standard form. Verify your equations using your calculator. A a. b. EVE
Write the equation of each function shown below in vertex and standard form. Verify your equations using your calculator. A a. b. EVE
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Educational Content on Quadratic Functions and Equations**
**Quadratic Functions**
4. Write the equation of each function shown below in vertex and standard form. Verify your equations using your calculator.
- **Graph a**: This graph depicts a parabola that opens upward, intersecting the y-axis at an approximate value of 0.
- **Graph b**: This graph shows a downward-opening parabola with its vertex on the negative side of the x-axis.
- **Graph c**: Here, the parabola opens downward with its vertex close to the y-axis.
**Level 4: Quadratic Revenue Model**
5. The equation \( R(x) = -0.5x^2 + 15x + 20 \) models the revenue (profit - costs) of a company in hundreds of thousands of dollars for every thousand units sold. Identify all key characteristics:
- **Vertex**: Indicates the maximum revenue.
- **X-intercepts**: Points where the company breaks even.
How many units can the company sell and still make a profit?
**Solving Quadratic Equations**
2. Solve the equation for the value(s) of \( x \). (There should be 2 answers for x value. Finding 1 is good enough for today’s ET, but if you find both x values and get 100 on today’s ET, you can change a lowest ET to 100 in Q1.)
\[ x^2 + 6x = 16 \]
**Graphical Explanation**
- **Visual Analysis**: Observe how the graphs showcase the properties of quadratic equations, including vertex, axis of symmetry, and intercepts. Graphical insights are significant in understanding solution methods and contextual problems like revenue models.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c6a3dff-6729-485c-933e-bf615eefedb6%2F8a7bfc77-20a0-41e6-81ec-c587f91b7266%2Fo4fvtzt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content on Quadratic Functions and Equations**
**Quadratic Functions**
4. Write the equation of each function shown below in vertex and standard form. Verify your equations using your calculator.
- **Graph a**: This graph depicts a parabola that opens upward, intersecting the y-axis at an approximate value of 0.
- **Graph b**: This graph shows a downward-opening parabola with its vertex on the negative side of the x-axis.
- **Graph c**: Here, the parabola opens downward with its vertex close to the y-axis.
**Level 4: Quadratic Revenue Model**
5. The equation \( R(x) = -0.5x^2 + 15x + 20 \) models the revenue (profit - costs) of a company in hundreds of thousands of dollars for every thousand units sold. Identify all key characteristics:
- **Vertex**: Indicates the maximum revenue.
- **X-intercepts**: Points where the company breaks even.
How many units can the company sell and still make a profit?
**Solving Quadratic Equations**
2. Solve the equation for the value(s) of \( x \). (There should be 2 answers for x value. Finding 1 is good enough for today’s ET, but if you find both x values and get 100 on today’s ET, you can change a lowest ET to 100 in Q1.)
\[ x^2 + 6x = 16 \]
**Graphical Explanation**
- **Visual Analysis**: Observe how the graphs showcase the properties of quadratic equations, including vertex, axis of symmetry, and intercepts. Graphical insights are significant in understanding solution methods and contextual problems like revenue models.
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