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Zorluk: KolayPolynomial Factors and Graphs

The graph of the polynomial function ff in the xyxy-plane is defined by f(x)=a(x−3)(x+4)f(x) = a(x - 3)(x + 4), where aa is a constant. If the yy-intercept of the graph is (0,−24)(0, -24), what is the value of aa?

  1. A
    -2
  2. B
    12\frac{1}{2}
  3. 2Cevap
  4. D
    -12

Cevap

2
The correct answer is 2. The yy-intercept of a graph is the point where x=0x = 0. For the function f(x)=a(x−3)(x+4)f(x) = a(x - 3)(x + 4), substituting x=0x = 0 yields f(0)=a(0−3)(0+4)=a(−3)(4)=−12af(0) = a(0 - 3)(0 + 4) = a(-3)(4) = -12a. Given that the yy-intercept is (0,−24)(0, -24), the value of the function at x=0x = 0 is −24-24. Setting these two values equal gives the equation −12a=−24-12a = -24. Dividing both sides of the equation by −12-12 isolates the constant, resulting in a=2a = 2.

Adım Adım Çözüm

1
Identify the relationship between the yy-intercept and the function value at x=0x = 0.
Since the yy-intercept is (0,−24)(0, -24), it follows that f(0)=−24f(0) = -24.
By definition, the yy-intercept of a graph is the point where the graph crosses the vertical axis, which occurs at x=0x = 0.
2
Substitute x=0x = 0 into the polynomial expression for f(x)f(x).
f(0)=a(0−3)(0+4)=a(−3)(4)=−12af(0) = a(0 - 3)(0 + 4) = a(-3)(4) = -12a.
This evaluates the algebraic expression at the yy-intercept to express the value in terms of the constant aa.
3
Equate the evaluated expression to the known yy-intercept value and solve for aa.
−12a=−24⇒a=2-12a = -24 \Rightarrow a = 2.
Setting the two expressions for f(0)f(0) equal allows us to solve the linear equation for the constant aa.

Anahtar Kavram

Evaluating a polynomial at x=0x = 0 to relate its algebraic form to its yy-intercept.
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Konular

Polynomial Factors and Graphs

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