What are the vertical asymptotes of the function f(x) = the quantity of 4 x plus 8, all over x squared plus 3 x minus 4? x = −1 and x = −2 x = −1 and x = 2 x = −1 and x = −4 x = 1 and x = −4

To find a rational function's vertical asymyptotes, you need to find when the denominator is zero. So when is the denominator x² + 3x - 4 zero? We set that denominator equal to zero.

x² + 3x - 4 = 0

This second degree equation factors. We want a pair of numbers that multiply to 4, and there are only two pairs: 2 and 2, 1 and 4. We want the sum to 3, and one must be negative. -2 and 2 add to zero, -4 and 1 add to -3, -1 and 4 add to 3. The last one is the pair.

x² + 3x - 4 = 0

(x + 4) (x - 1) = 0

x = -4 or x = 1

So the function's vertical asymptotes are at x = 1 and x = -4.

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.