Math Problem Statement

a parabola passes through the points (04), (1,0), and (-2,0). find the equation of the parabola

Solution

To find the equation of the parabola that passes through the points (0,4)(0, 4), (1,0)(1, 0), and (2,0)(-2, 0), we will assume that the equation of the parabola is of the form:

y=ax2+bx+cy = ax^2 + bx + c

Where aa, bb, and cc are constants we need to determine. Given that the parabola passes through the points, we can substitute the coordinates of the points into this equation and create a system of linear equations to solve for aa, bb, and cc.

Step 1: Use the point (0,4)(0, 4)

Substitute x=0x = 0 and y=4y = 4 into the general form of the parabola:

4=a(0)2+b(0)+cc=44 = a(0)^2 + b(0) + c \quad \Rightarrow \quad c = 4

So, the equation simplifies to:

y=ax2+bx+4y = ax^2 + bx + 4

Step 2: Use the point (1,0)(1, 0)

Substitute x=1x = 1 and y=0y = 0 into the equation:

0=a(1)2+b(1)+4a+b+4=00 = a(1)^2 + b(1) + 4 \quad \Rightarrow \quad a + b + 4 = 0 a+b=4(Equation 1)a + b = -4 \quad \text{(Equation 1)}

Step 3: Use the point (2,0)(-2, 0)

Substitute x=2x = -2 and y=0y = 0 into the equation:

0=a(2)2+b(2)+44a2b+4=00 = a(-2)^2 + b(-2) + 4 \quad \Rightarrow \quad 4a - 2b + 4 = 0 4a2b=4(Equation 2)4a - 2b = -4 \quad \text{(Equation 2)}

Step 4: Solve the system of equations

Now, we have the following system of equations:

  1. a+b=4a + b = -4
  2. 4a2b=44a - 2b = -4

Solve Equation 1 for bb:

b=4ab = -4 - a

Substitute into Equation 2:

4a2(4a)=44a - 2(-4 - a) = -4 4a+8+2a=44a + 8 + 2a = -4 6a+8=46a + 8 = -4 6a=126a = -12 a=2a = -2

Substitute a=2a = -2 into b=4ab = -4 - a:

b=4(2)=4+2=2b = -4 - (-2) = -4 + 2 = -2

Step 5: Write the final equation

Now that we have a=2a = -2, b=2b = -2, and c=4c = 4, the equation of the parabola is:

y=2x22x+4y = -2x^2 - 2x + 4

Final Answer:

The equation of the parabola is:

y=2x22x+4y = -2x^2 - 2x + 4

Would you like more details on how to solve this, or do you have other questions?

Further Questions:

  1. What is the vertex of the parabola y=2x22x+4y = -2x^2 - 2x + 4?
  2. How can we find the axis of symmetry for this parabola?
  3. How do you determine the direction (upward or downward) of the parabola from the equation?
  4. How do you find the focus and directrix of this parabola?
  5. How can you use the quadratic formula to find the roots of the parabola equation?

Tip:

To find the vertex of a parabola given its equation y=ax2+bx+cy = ax^2 + bx + c, use the formula for the xx-coordinate of the vertex: x=b2ax = -\frac{b}{2a}.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Systems of Equations

Formulas

y = ax^2 + bx + c
System of linear equations

Theorems

Substitution method for solving systems of equations

Suitable Grade Level

Grades 9-11