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How Do You Write a Quadratic Equation in Vertex Form if You Have the Vertex and Another Point?

Write an equation in vertex form for the parabola that has a vertex of (-6,-4) and passes through the point (-2,12).

Summary

  1. Vertex form of a quadratic equation is y=a(x-h)2+k, where (h,k) is the vertex of the parabola
  2. The vertex of a parabola is the point at the top or bottom of the parabola
  3. 'h' is -6, the first coordinate in the vertex
  4. 'k' is -4, the second coordinate in the vertex
  5. 'x' is -2, the first coordinate in the other point
  6. 'y' is 12, the second coordinate in the other point

Notes

    1. Vertex form of a quadratic equation is y=a(x-h)2+k, where (h,k) is the vertex of the parabola
    1. Vertex form is useful, because it lets us pick out the vertex of a parabola really quickly just by looking at the equation!
    2. Notice how there is a minus sign in front of 'h' in vertex form
    3. So we'll need to be careful when we pick out 'h', because we'll have to switch the sign that's in front of it in the equation
    1. Once we figure out what we've been given, we'll be able to tell what we still need to find
    1. The vertex of a parabola is the point at the top or bottom of the parabola
    2. Remember, in vertex form the vertex is the point (h,k)
    3. So the first coordinate in the vertex, -6, will be 'h' in our equation
    4. The second coordinate in the vertex, -4, will be 'k' in our equation
    1. We know that our parabola also goes through the point (-2,12)
    2. Remember, ordered pairs are of the form (x,y)
    3. Since we know (-2,12) is on our graph, we can use those values for 'x' and 'y' to help us find our equation
    4. The first coordinate in an ordered pair is the x-coordinate, so x will be -2
    5. The second coordinate in an ordered pair is the y-coordinate, so y will be 12
    1. We have everything we need to write our equation in vertex form except for 'a'
    2. But if we plug everything else into the vertex form equation, we'll be able to solve for 'a'
    3. Remember, vertex form is y=a(x-h)2+k
    4. We know that h = -6, k = -4, x = -2, and y = 12
    5. Since our value for 'k' is negative, we'll subtract it instead of adding it
    1. The only thing we're missing in our equation is 'a'
    2. So we can simplify and solve our equation to find 'a'
    1. First we need to simplify inside the parentheses by subtracting -6 from -2 to get 4
    2. Then we square 4 to get 16
    3. Add 4 to both sides to get the 'a' term by itself
    4. Then divide both sides by 16: the 16's cancel out on both sides, leaving us with a=1
    1. Now that we have values for 'a', 'h', and 'k', we can write a general equation for our parabola using vertex form
    2. Remember, vertex form is y=a(x-h)2+k
    1. Remember, vertex form is y=a(x-h)2+k
    2. To write a general equation for our parabola, we want to leave 'x' and 'y' as variables but have values for everything else
    3. We found that a=1, h=-6, and k=-4, so we can plug these values into the general form to get our equation
    4. So we get y=(x-(-6))2-4 when we plug in
    5. Simplifying inside the parentheses we get y=(x+6)2-4 for our final equation
    1. Remember, vertex form is y=a(x-h)2+k
    2. To write a general equation for our parabola, we want to leave 'x' and 'y' as variables but have values for everything else
    3. We found that a=1, h=-6, and k=-4, so we can plug these values into the general form to get our equation
    4. So we get y=(x-(-6))2-4 when we plug in
    5. Simplifying inside the parentheses we get y=(x+6)2-4 for our final equation