Representing Situations with Tables, Graphs, and Equations in y = kx or y = x + b Form(TEKS.Math.6.6.C)
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Texas 6th Grade Math › Representing Situations with Tables, Graphs, and Equations in y = kx or y = x + b Form(TEKS.Math.6.6.C)
Situation: A dog-walking service charges a \$5 booking fee plus \$12 per hour. Which equation represents this situation, where x is the number of hours and y is the total cost?
Equation: $y = 12x + 5$
Table: x|y -> 0|12; 1|24; 2|36; 3|48
Graph description: a straight line passing through the origin with slope 12
Verbal: The total cost is a \$12 fee plus \$5 for each hour
Explanation
The \$5 is a fixed fee (y-intercept), and \$12 per hour is the rate (slope). So $y = 12x + 5$. The table and graph that start at $y=0$ ignore the fixed fee, and the swapped-fee verbal description is incorrect.
Situation: A canoe rental costs \$8 per hour with no sign-up fee. Which table shows the relationship between hours x and total cost y?
Equation: $y = 8x + 10$
Table: x|y -> 0|0; 1|8; 2|16; 3|24
Graph description: a line crossing the y-axis at 8 and rising very slowly (slope 1/8)
Verbal: It costs \$8 to start plus \$1 each hour
Explanation
This is proportional with no fixed fee, so $y = 8x$. The correct table shows $(0,0)$ and increases by 8 each hour. The equation with +10 and the verbal with a start fee are wrong, and the graph description mentions a y-intercept of 8, which is not proportional.
Situation: A streaming plan charges a \$10 sign-up fee plus \$6 per month. Which graph description best represents this situation?
Equation: $y = 10x + 6$
Table: x|y -> 0|0; 1|6; 2|12; 3|18
Graph description: a straight line that crosses the y-axis at 10 and increases by 6 for each 1 step to the right
Verbal: The plan costs \$6 to start and then \$10 each month
Explanation
With a $10$ sign-up fee and $6$ per month, the equation is $y = 6x + 10$. Its graph has y-intercept 10 and slope 6. The given table ignores the fixed fee, the equation swaps the fee and rate, and the verbal description reverses them.
Situation: A ride-share charges a \$2 base fare plus \$1.20 per mile. Which equation represents this situation, where x is miles and y is total cost?
Table: x|y -> 0|0.00; 1|1.20; 3|3.60; 5|6.00
Graph description: a line through the origin with slope 1.2
Verbal: You pay \$1.20 to begin the ride and then \$2 per mile after that
Equation: $y = 1.20x + 2$
Explanation
The base fare is the y-intercept ($2$) and the per-mile rate is the slope (\$1.20$), so $y = 1.20x + 2$. The table and graph through the origin ignore the base fare, and the verbal description swaps the base and rate.
Situation: For a school fundraiser, you earn \$15 per ticket sold, plus a one-time \$20 bonus. Which verbal description correctly matches this relationship?
Verbal: The total money $y$ is $20$ to start plus $15$ for each ticket $x$ sold
Graph description: a line crossing the x-axis at 20 with slope 15
Table: x|y -> 0|0; 1|15; 2|30; 3|45
Equation: $y = 15x - 20$
Explanation
The relationship is $y = 15x + 20$: $20$ is the fixed bonus (y-intercept) and $15$ per ticket is the rate (slope). The graph description confuses intercepts, the table omits the bonus, and the equation has the wrong sign for the fixed amount.