Math Problem Statement
Consider the following function. Complete parts (a) through (e) below. y equals x squared plus 8 x plus 15 Question content area bottom left Part 1 a. Determine if the parabola whose equation is given opens upward or downward. Choose the correct answer below. The parabola opens downward. The parabola opens upward. Your answer is correct. Part 2 b. Find the vertex. The vertex is left parenthesis negative 4 comma negative 1 right parenthesis. (Type an ordered pair.) Part 3 c. Find the x-intercepts. Select the correct answer and, if necessary, fill in the answer box to complete your choice. A. The x-intercept(s) is/are enter your response here. (Type an integer or a fraction. Use a comma to separate answers as needed.) B. There are no x-intercepts.
Solution
We are given the quadratic function . Let's go through the different parts step by step:
Part 1: Opening of the Parabola
The general form of a quadratic equation is . The sign of the coefficient determines whether the parabola opens upward or downward.
- If , the parabola opens upward.
- If , the parabola opens downward.
In this case, , which is positive. So, the parabola opens upward.
Part 2: Vertex of the Parabola
The vertex of a parabola in the form can be found using the formula for the x-coordinate of the vertex:
For the given equation , we have and . Plugging these values into the formula:
To find the y-coordinate of the vertex, substitute into the equation :
Thus, the vertex is .
Part 3: Finding the x-intercepts
The x-intercepts occur when . We solve the equation .
This is a quadratic equation, which can be factored:
Setting each factor equal to zero:
Solving these equations gives:
Thus, the x-intercepts are and , or as an ordered pair: and .
Summary of Answers:
- Part a: The parabola opens upward.
- Part b: The vertex is .
- Part c: The x-intercepts are and .
Would you like more details on any part of this process or have any other questions?
Here are 5 related questions for further practice:
- What is the y-intercept of the function ?
- How do you find the axis of symmetry for a parabola?
- Can a quadratic function have more than two x-intercepts?
- How would the graph of the function change if the coefficient of were negative?
- How do you complete the square to rewrite the function in vertex form?
Tip: Factoring is a quick way to find x-intercepts when the quadratic is easily factorable, but using the quadratic formula works for all cases!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Parabolas
Formulas
Quadratic equation formula: y = ax^2 + bx + c
Vertex formula: x = -b / 2a
Factoring method for finding x-intercepts
Theorems
Properties of parabolas based on the sign of 'a'
Factoring quadratics
Suitable Grade Level
Grades 9-10