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Zorluk: OrtaLinear Functions and Graphs

The graph of the linear function ff in the xyxy-plane has a yy-intercept of (0,b)(0, b) and an xx-intercept of (a,0)(a, 0), where aa and bb are nonzero constants. If 3a=−4b3a = -4b, which of the following is the slope of the graph of ff?

  1. A
    −43-\frac{4}{3}
  2. B
    −34-\frac{3}{4}
  3. 34\frac{3}{4}Cevap
  4. D
    43\frac{4}{3}

Cevap

34\frac{3}{4}
The slope of the line passing through (0,b)(0, b) and (a,0)(a, 0) is m=0−ba−0=−bam = \frac{0 - b}{a - 0} = -\frac{b}{a}. Starting with the given equation 3a=−4b3a = -4b, dividing both sides by −4a-4a isolates the slope expression: −ba=34-\frac{b}{a} = \frac{3}{4}. Therefore, the slope of the graph of ff is 34\frac{3}{4}.

Adım Adım Çözüm

1
Identify the coordinates of the intercepts and write the formula for the slope of a line.
The yy-intercept is (0,b)(0, b) and the xx-intercept is (a,0)(a, 0). The slope mm is given by m=0−ba−0=−bam = \frac{0 - b}{a - 0} = -\frac{b}{a}.
To find the slope, we express it in terms of the variables aa and bb using the standard slope formula.
2
Use the given equation to find the ratio −ba-\frac{b}{a}.
Divide both sides of the equation 3a=−4b3a = -4b by aa to get 3=−4(ba)3 = -4\left(\frac{b}{a}\right). Then, divide both sides by −4-4 to get −34=ba-\frac{3}{4} = \frac{b}{a}, which means −ba=34-\frac{b}{a} = \frac{3}{4}.
We isolate the expression for the slope, which is −ba-\frac{b}{a}, using algebraic operations on the given equation.
3
Equate the slope expression to the calculated value.
Since m=−bam = -\frac{b}{a} and −ba=34-\frac{b}{a} = \frac{3}{4}, the slope of the line is 34\frac{3}{4}.
This yields the final value of the slope.

Anahtar Kavram

Calculating the slope of a linear function using intercepts and algebraic substitution.
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Konular

Linear Functions and Graphs

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