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Magazine Chapter 11: Problem 48
Use a graphing calculator to find the vertex of the graph of each function. $$ f(x)=0.5 x^{2}+2.4 x+3.2 $$
Short Answer
Expert verified
(-2.4, 0.32)
Step by step solution
01
Identify the quadratic function
Recognize that the function given is a quadratic function of the form y = ax^2 + bx + c In this case, a = 0.5, b = 2.4, c = 3.2.
02
Use vertex formula
The vertex (h, k) of a quadratic function in standard form can be found using the formula:\[ h = -\frac{b}{2a} \]Substitute a = 0.5 and b = 2.4 into the vertex formula.
03
Calculate the x-coordinate of the vertex
Using the values:\[ h = -\frac{2.4}{2 \times 0.5} \]\[ h = -\frac{2.4}{1} \]\[ h = -2.4 \]Thus, the x-coordinate of the vertex is -2.4.
04
Calculate the y-coordinate of the vertex
Substitute x = -2.4 into the original function to find the y-coordinate:\[ f(-2.4) = 0.5(-2.4)^2 + 2.4(-2.4) + 3.2 \]Calculate step-by-step:\[0.5(-2.4)^2 = 0.5 \times 5.76 = 2.88\]\[2.4(-2.4) = -5.76\]Then,\[ f(-2.4) = 2.88 - 5.76 + 3.2 \]\[ f(-2.4) = 2.88 - 5.76 + 3.2 = 0.32 \]Thus, the y-coordinate of the vertex is 0.32.
05
Write the vertex
The vertex of the function is \[ (-2.4, 0.32) \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
graphing calculator
A graphing calculator is a powerful tool that can help you visualize and solve mathematical problems, especially when dealing with quadratic functions. With a graphing calculator, you can easily plot the graph of a function and identify key features like the vertex, roots, and axis of symmetry. To find the vertex of a quadratic function using a graphing calculator, you would:
- Enter the function into the calculator.
- Graph the function to see its parabolic shape.
- Use the calculator's 'vertex' or 'minimum/maximum' function options to automatically find the vertex coordinates.
quadratic function
A quadratic function is a polynomial function of degree two, and its general form is written as \( y = ax^2 + bx + c \). In this case, a, b, and c are constants, and the graph of a quadratic function is always a parabola. There are some important aspects to consider when studying quadratic functions:
- Parabola Direction: If the coefficient 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.
- Vertex: The vertex is the highest or lowest point on the graph, depending on the parabola's direction.
- Axis of Symmetry: The parabola is symmetric about a vertical line that passes through the vertex. This line is called the axis of symmetry.
- Roots: Also known as x-intercepts or zeros, these are the points where the graph crosses the x-axis.
vertex formula
The vertex formula is used to find the vertex of a quadratic function written in the standard form \( y = ax^2 + bx + c \). The formula to calculate the x-coordinate (h) of the vertex is \[ h = -\frac{b}{2a} \]. Once you have the x-coordinate, you can substitute it back into the original quadratic function to find the y-coordinate (k), giving you the vertex (h, k). Let's walk through the process with the example function \( f(x)=0.5x^2+2.4x+3.2 \):
- First, identify the values of a, b, and c. Here, \( a=0.5 \), \( b=2.4 \), and \( c=3.2 \).
- Plug the values into the formula: \[ h = -\frac{2.4}{2 \times 0.5} = -2.4 \]
- Use the x-coordinate to find the y-coordinate: \[ f(-2.4) = 0.5(-2.4)^2 + 2.4(-2.4) + 3.2 = 0.32 \]
standard form
The standard form of a quadratic function is one of the essential ways to represent a quadratic equation as \( y = ax^2 + bx + c \). Here:
- 'a' determines the parabola's direction and width.
- 'b' affects the position of the vertex along the x-axis.
- 'c' represents the y-intercept, where the graph crosses the y-axis.
- Easy Identification: You can quickly identify whether the parabola opens upwards or downwards by looking at the sign of 'a'.
- Vertex Calculation: You can easily use the vertex formula \( h = -\frac{b}{2a} \) to find the vertex.
- Factorization: It sets the stage for converting the equation into factored form or completing the square to get the vertex form.
