Math Problem Statement
Solution
Let's break down the three questions from the image you uploaded:
Question 1:
We are given the function , and we need to determine which of the points lie on this parabola. For each point, substitute the -value into the function and see if the corresponding -value matches.
-
Point :
- Substituting into :
- Since , the point does not lie on the parabola.
-
Point :
- Substituting into :
- Since , the point does lie on the parabola.
-
Point :
- Substituting into :
- Since , the point does not lie on the parabola.
-
Point :
- Substituting into :
- Since , the point does not lie on the parabola.
Answer: The point lies on the parabola.
Question 2:
We are given three points and need to determine which quadratic equation passes through these points. We can use the general form of a quadratic function and substitute the points to find the equation.
- Points:
Let's substitute each point into the general quadratic form and solve for , , and .
- Substituting :
- Substituting :
- Substituting :
Solving these three equations will give the values of , , and , allowing us to determine the correct quadratic equation.
Question 3:
We are given the function which models the height of a baseball at time , and we need to find the maximum height of the ball.
Since the quadratic function has a negative coefficient for , it opens downward, indicating that the vertex gives the maximum height. The formula for the time of the vertex of a parabola is: Here, and , so the time at which the maximum height occurs is:
Substitute into the equation for to find the maximum height:
Answer: The maximum height is 1.8 meters.
Would you like more details or have any further questions about these problems? Here are some related questions:
- How do we solve for , , and in a system of quadratic equations?
- What is the vertex form of a quadratic equation?
- How can we solve for the maximum or minimum of a quadratic function?
- How do we calculate the height of an object at any given time from a quadratic function?
- How can we graph a quadratic equation using its vertex and other points?
Tip: When solving for points on a parabola, always substitute the -value into the given equation and compare the result to the -value of the point to check if it lies on the curve.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Parabolas
Vertex Form
Systems of Equations
Formulas
Standard quadratic equation y = ax^2 + bx + c
Vertex formula t = -b/(2a)
Substitution to verify points on a curve
Theorems
Vertex Theorem for Parabolas
Suitable Grade Level
Grades 9-11