Math Problem Statement

Which of the following points lie on the parabola f(x) = -x^2 + 5x + 3? Which quadratic function passes through the points (-1, 5), (4, 0), and (5, -7)? What is the maximum height the baseball can reach modeled by h(t) = -5t^2 + 4t + 1?

Solution

Let's break down the three questions from the image you uploaded:

Question 1:

We are given the function f(x)=x2+5x+3f(x) = -x^2 + 5x + 3, and we need to determine which of the points lie on this parabola. For each point, substitute the xx-value into the function and see if the corresponding yy-value matches.

  • Point (1,3)(1, 3):

    • Substituting x=1x = 1 into f(x)f(x): f(1)=(1)2+5(1)+3=1+5+3=7f(1) = -(1)^2 + 5(1) + 3 = -1 + 5 + 3 = 7
    • Since f(1)=7f(1) = 7, the point (1,3)(1, 3) does not lie on the parabola.
  • Point (1,7)(1, 7):

    • Substituting x=1x = 1 into f(x)f(x): f(1)=(1)2+5(1)+3=7f(1) = -(1)^2 + 5(1) + 3 = 7
    • Since f(1)=7f(1) = 7, the point (1,7)(1, 7) does lie on the parabola.
  • Point (1,8)(1, 8):

    • Substituting x=1x = 1 into f(x)f(x): f(1)=(1)2+5(1)+3=7f(1) = -(1)^2 + 5(1) + 3 = 7
    • Since f(1)=7f(1) = 7, the point (1,8)(1, 8) does not lie on the parabola.
  • Point (1,9)(1, 9):

    • Substituting x=1x = 1 into f(x)f(x): f(1)=(1)2+5(1)+3=7f(1) = -(1)^2 + 5(1) + 3 = 7
    • Since f(1)=7f(1) = 7, the point (1,9)(1, 9) does not lie on the parabola.

Answer: The point (1,7)(1, 7) 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 y=ax2+bx+cy = ax^2 + bx + c and substitute the points to find the equation.

  • Points: (1,5),(4,0),(5,7)(-1, 5), (4, 0), (5, -7)

Let's substitute each point into the general quadratic form and solve for aa, bb, and cc.

  1. Substituting (1,5)(-1, 5): 5=a(1)2+b(1)+c    5=ab+c(Equation 1)5 = a(-1)^2 + b(-1) + c \implies 5 = a - b + c \quad \text{(Equation 1)}
  2. Substituting (4,0)(4, 0): 0=a(4)2+b(4)+c    0=16a+4b+c(Equation 2)0 = a(4)^2 + b(4) + c \implies 0 = 16a + 4b + c \quad \text{(Equation 2)}
  3. Substituting (5,7)(5, -7): 7=a(5)2+b(5)+c    7=25a+5b+c(Equation 3)-7 = a(5)^2 + b(5) + c \implies -7 = 25a + 5b + c \quad \text{(Equation 3)}

Solving these three equations will give the values of aa, bb, and cc, allowing us to determine the correct quadratic equation.

Question 3:

We are given the function h(t)=5t2+4t+1h(t) = -5t^2 + 4t + 1 which models the height of a baseball at time tt, and we need to find the maximum height of the ball.

Since the quadratic function has a negative coefficient for t2t^2, it opens downward, indicating that the vertex gives the maximum height. The formula for the time tt of the vertex of a parabola y=at2+bt+cy = at^2 + bt + c is: t=b2at = \frac{-b}{2a} Here, a=5a = -5 and b=4b = 4, so the time at which the maximum height occurs is: t=42(5)=410=0.4secondst = \frac{-4}{2(-5)} = \frac{4}{10} = 0.4 \, \text{seconds}

Substitute t=0.4t = 0.4 into the equation for h(t)h(t) to find the maximum height: h(0.4)=5(0.4)2+4(0.4)+1=5(0.16)+1.6+1=0.8+1.6+1=1.8metersh(0.4) = -5(0.4)^2 + 4(0.4) + 1 = -5(0.16) + 1.6 + 1 = -0.8 + 1.6 + 1 = 1.8 \, \text{meters}

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:

  1. How do we solve for aa, bb, and cc in a system of quadratic equations?
  2. What is the vertex form of a quadratic equation?
  3. How can we solve for the maximum or minimum of a quadratic function?
  4. How do we calculate the height of an object at any given time from a quadratic function?
  5. How can we graph a quadratic equation using its vertex and other points?

Tip: When solving for points on a parabola, always substitute the xx-value into the given equation and compare the result to the yy-value of the point to check if it lies on the curve.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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