Mathematics

Question

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar.

Determine the equation for the parabola graphed below.
y =
x2 +
x +

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar. Determine the equation for the parabola graphed be

1 Answer

  • Answer:

    [tex]\large\boxed{y=\dfrac{1}{2}x^2-2x+1}[/tex]

    Step-by-step explanation:

    The vertex form of an equation of a parabola:

    [tex]y=a(x-h)^2+k[/tex]

    (h, k) - vertex

    a - leading coefficient in equation y = ax² + bx + c

    From the grap we can read coordinates of the vertex (2, -1) and y-intercept (0, 1).

    Therefore h = 2, k = -1

    Put the values of h, k and coordinates of the y-intercept to the equation of parabola:

    [tex]1=a(0-2)^2-1[/tex]     add 1 to both sides

    [tex]1+1=a(2)^2-1+1[/tex]

    [tex]2=4a[/tex]              divide both sides by 4

    [tex]\dfrac{2}{4}=\dfrac{4a}{4}\\\\\dfrac{1}{2}=a\to a=\dfrac{1}{2}[/tex]

    Therefore we have the equation:

    [tex]y=\dfrac{1}{2}(x-2)^2-1[/tex]

    Convert to the standard form:

    [tex]y=\dfrac{1}{2}(x-2)^2-1[/tex]       use (a - b)² = a² - 2ab + b²

    [tex]y=\dfrac{1}{2}(x^2-2(x)(2)+2^2)-1[/tex]

    [tex]y=\dfrac{1}{2}(x^2-4x+4)-1[/tex]           use the distributive property

    [tex]y=\dfrac{1}{2}x^2-\dfrac{1}{2}\cdot4x+\dfrac{1}{2}\cdot4-1[/tex]

    [tex]y=\dfrac{1}{2}x^2-2x+2-1[/tex]          combine like terms

    [tex]y=\dfrac{1}{2}x^2-2x+1[/tex]

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