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Math  /  Algebra

QuestionGiven the points on the x-axis: (−2,0) and (4,0), and a point on the parabola: (5,2), find the y-intercept (y-axis intercept) of the parabola.\text{Given the points on the x-axis: } (-2, 0) \text{ and } (4, 0), \text{ and a point on the parabola: } (5, 2), \text{ find the y-intercept (} y \text{-axis intercept) of the parabola.}

Studdy Solution

STEP 1

1. The parabola is in the standard quadratic form y=ax2+bx+c y = ax^2 + bx + c .
2. The points on the x-axis, (−2,0) (-2, 0) and (4,0) (4, 0) , are the roots of the parabola.
3. The point (5,2) (5, 2) lies on the parabola.
4. The y-intercept of the parabola is the value of y y when x=0 x = 0 .

STEP 2

1. Use the roots to write the quadratic equation in factored form.
2. Use the given point to find the value of a a .
3. Calculate the y-intercept by evaluating the equation at x=0 x = 0 .

STEP 3

Since the roots of the parabola are x=−2 x = -2 and x=4 x = 4 , we can write the equation of the parabola in factored form:
y=a(x+2)(x−4) y = a(x + 2)(x - 4)

STEP 4

Substitute the point (5,2) (5, 2) into the equation to find a a . This point satisfies the equation:
2=a(5+2)(5−4) 2 = a(5 + 2)(5 - 4)

STEP 5

Simplify and solve for a a :
2=a(7)(1) 2 = a(7)(1) 2=7a 2 = 7a a=27 a = \frac{2}{7}

STEP 6

Substitute a=27 a = \frac{2}{7} back into the factored form of the equation:
y=27(x+2)(x−4) y = \frac{2}{7}(x + 2)(x - 4)

STEP 7

To find the y-intercept, substitute x=0 x = 0 into the equation:
y=27(0+2)(0−4) y = \frac{2}{7}(0 + 2)(0 - 4)

STEP 8

Simplify to find the y-intercept:
y=27(2)(−4) y = \frac{2}{7}(2)(-4) y=27×−8 y = \frac{2}{7} \times -8 y=−167 y = -\frac{16}{7}
The y-intercept of the parabola is:
−167 \boxed{-\frac{16}{7}}

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