Math Problem Statement
Solution
This image shows a math problem related to quadratic functions, where three equivalent quadratic equations are given, and the task is to evaluate different statements about the graph of the function.
The three quadratic equations are:
And the four statements to evaluate are:
a) The graph has a y-intercept at .
b) The coordinates of the vertex are .
c) The graph of the function opens upward.
d) The x-intercepts are at and .
Let me break down each part to help evaluate whether these statements are true or false.
a) The graph has a y-intercept at .
To check this, we substitute in each of the equations to find the y-intercept.
For equation A, : Thus, the y-intercept is , not . So, statement a is false.
b) The coordinates of the vertex are .
To find the vertex, we need to express the equation in vertex form. Let's check each equation.
For equation A, , we complete the square to find the vertex form: This is the vertex form, and the vertex is indeed . So, statement b is true.
c) The graph of the function opens upward.
A quadratic function opens upward if the coefficient of is positive. In all the given equations, the coefficient of is positive. Therefore, the graph opens upward. So, statement c is true.
d) The x-intercepts are at and .
To find the x-intercepts, we set and solve for .
For equation A, , we solve: Factoring gives: Thus, the x-intercepts are and , not 3 and -5. Therefore, statement d is false.
Final Answers:
- a) False
- b) True
- c) True
- d) False
Would you like me to explain any of these steps in more detail, or do you have any further questions about this problem?
Related Questions:
- How do you complete the square to convert a quadratic equation into vertex form?
- What does the vertex form of a quadratic function tell you about the graph?
- How do you determine whether a parabola opens upward or downward from the equation?
- How can you factor a quadratic equation to find the x-intercepts?
- How does the discriminant of a quadratic equation help you determine the number of real x-intercepts?
Tip:
When finding the vertex of a quadratic equation, always check if the equation is in standard form and then try to complete the square or use the vertex formula to directly find the x-coordinate of the vertex.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Vertex Form
Factoring Quadratics
Formulas
Standard form of a quadratic equation: f(x) = ax^2 + bx + c
Vertex form of a quadratic equation: f(x) = a(x-h)^2 + k
Factoring: f(x) = (x-r1)(x-r2)
Theorems
Properties of parabolas: Opening direction determined by coefficient of x^2
Vertex of a parabola at x = -b/(2a)
Suitable Grade Level
Grade 9-10
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