Graphing Quadratic Functions: Chapter 10 Review (2024)

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8 Questions

What is the vertex form of a quadratic function, and how do you use it to graph the function?

The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex. To graph the function, you can plot the vertex and 4 other points, and then connect the points to form a parabola.

What is the quadratic formula, and how is it used to solve quadratic equations?

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation. It is used to solve quadratic equations by providing the solutions to the equation.

How do you find the vertex of a quadratic function in the form y = ax^2 + bx + c?

To find the vertex of a quadratic function in the form y = ax^2 + bx + c, you can use the formula x = -b / 2a. This will give you the x-coordinate of the vertex, and you can then substitute this value into the function to find the y-coordinate.

What is the relationship between the coefficient of the x^2 term and the shape of the parabola?

The coefficient of the x^2 term determines the direction of the parabola's opening. If the coefficient is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards.

What is the meaning of the productivity equation P = –0.025t^2 + 3t, in the context of the word problem?

The productivity equation P = –0.025t^2 + 3t represents the relationship between the length of a class (in minutes) and the productivity of the students (in %).

What is the productivity of a class that is 90 minutes long, according to the equation P = –0.025t^2 + 3t?

To find the productivity, substitute t = 90 into the equation: P = –0.025(90)^2 + 3(90) = 67.5%. The productivity of a class that is 90 minutes long is approximately 67.5%.

What is the axis of symmetry of a quadratic function, and how is it related to the vertex?

The axis of symmetry is the vertical line that passes through the vertex of the parabola, and is given by the equation x = -b / 2a. The axis of symmetry is related to the vertex because it passes through the vertex and is perpendicular to the x-axis.

How do you determine the x-intercepts of a quadratic function, and what do they represent?

To find the x-intercepts, set the function equal to zero and solve for x. The x-intercepts represent the points at which the parabola intersects the x-axis, and are the solutions to the equation.

Study Notes

Quadratic Functions

Graph quadratic functions by showing the vertex and 4 other points:
- Example: 𝑦 =− (𝑥 − 2)² + 3
- Example: 𝑦 = 2(𝑥 + 1)² − 4
- Example: 𝑦= 2/(𝑥 +3)
- Example: 𝑦 = (𝑥 + 3)²
- Example: 𝑦 = 𝑥² − 4𝑥 + 1
- Example: 𝑦 =− 2/(𝑥 + 2𝑥)
- Example: 𝑦 =− 3/(𝑥 +3)²
- Example: 𝑦 = − 2𝑥² + 4𝑥 + 3

Solving Quadratic Equations

Use the quadratic formula to solve equations:
- Example: ½x² - x - 4 = 0
- Example: 4x² - 6x + 3 = 0
- Example: 4x² + 8x = 1
- Example: 3x² - 7x = -3

Word Problem: Productivity and Class Time

The relationship between class time and productivity is given by the equation: P = –0.025t² + 3t
- Where P is the productivity (in %) and t is the length of the class (in minutes)
The problem asks to find the productivity of a class that is 90 minutes long.

Review quiz on graphing quadratic functions, including identifying the vertex and plotting points. Covers equations in vertex form and standard form.