Graphing Linear Functions - Pre Algebra

1. A linear function has a y-intercept of $-3$ and passes through the point $(4, 5)$. If this function is graphed on a coordinate plane, what is the x-intercept?
   Answer: $\left(\frac{3}{2}, 0\right)$ First, find the slope using the y-intercept $(0, -3)$ and the given point $(4, 5)$: $m = \frac{5 - (-3)}{4 - 0} = \frac{8}{4} = 2$. The equation is $y = 2x - 3$. To find the x-intercept, set $y = 0$: $0 = 2x - 3$, so $x = \frac{3}{2}$. Choice B incorrectly uses the x-coordinate of the given point in the calculation. Choice C results from using the wrong slope formula. Choice D comes from setting $x = 3 \div 2$ incorrectly.
2. Two linear functions are graphed on the same coordinate plane. Function $f$ has equation $y = 2x - 1$ and function $g$ passes through points $(1, 3)$ and $(3, 7)$. How do their rates of change compare?
   Answer: Function $g$ has the same rate of change as function $f$ Function $f$ has rate of change (slope) $m_f = 2$. For function $g$, the rate of change is $m_g = \frac{7 - 3}{3 - 1} = \frac{4}{2} = 2$. Both functions have the same rate of change. Choice B incorrectly calculates the slope of $g$ as $4$. Choice C incorrectly calculates the slope of $g$ as $1$. Choice D confuses rate of change with the difference in y-values.
3. A linear function has the equation $y = mx + b$ where $m < 0$ and $b > 0$. When this function is graphed, which quadrants will the line definitely pass through?
   Answer: Quadrants I, II, and IV only When you see a linear function question asking about quadrants, you need to analyze how the slope and y-intercept affect the line's path through the coordinate plane. Given $y = mx + b$ where $m < 0$ and $b > 0$, let's trace what happens. Since $b > 0$, the y-intercept is positive, meaning the line crosses the y-axis above the origin. Since $m < 0$, the slope is negative, so the line falls from left to right. Starting from the positive y-intercept and moving in both directions: as $x$ increases (moving right), $y$ decreases due to the negative slope. Eventually, the line will cross the x-axis and enter Quadrant IV. As $x$ decreases (moving left from the y-intercept), $y$ increases, taking the line into Quadrant II. So the line definitely passes through Quadrants II (upper left), I (upper right, near the y-intercept), and IV (lower right). The line will never enter Quadrant III because it would need to have negative x-values AND negative y-values simultaneously, which is impossible when starting from a positive y-intercept with a finite negative slope. Choice A is wrong because it excludes Quadrant I, but the line passes through I near the y-intercept. Choice B incorrectly includes Quadrant III, which the line cannot reach. Choice D is wrong because, as explained, Quadrant III is impossible to reach. Study tip: For any linear function, always start by plotting the y-intercept, then use the slope to determine the line's direction. This visual approach makes quadrant questions much clearer.
4. A linear function is defined by $f(x) = -\frac{2}{3}x + 4$. If the graph of this function is drawn, which statement about its intercepts is correct?
   Answer: The y-intercept is $(0, 4)$ and the x-intercept is $(6, 0)$ When you encounter questions about intercepts of linear functions, you need to find where the graph crosses each axis. The y-intercept occurs where the line crosses the y-axis (when x = 0), and the x-intercept occurs where the line crosses the x-axis (when y = 0). For $f(x) = -\frac{2}{3}x + 4$, let's find both intercepts systematically. To find the y-intercept, substitute x = 0: $f(0) = -\frac{2}{3}(0) + 4 = 4$. So the y-intercept is $(0, 4)$. To find the x-intercept, set the function equal to 0 and solve for x: $0 = -\frac{2}{3}x + 4$. Subtract 4 from both sides: $-4 = -\frac{2}{3}x$. Multiply both sides by $-\frac{3}{2}$: $x = (-4) \times (-\frac{3}{2}) = 6$. So the x-intercept is $(6, 0)$. Choice C correctly identifies both intercepts as $(0, 4)$ and $(6, 0)$. Choice A has the correct y-intercept but places the x-intercept at $(-6, 0)$, which results from forgetting to multiply by the negative when solving for x. Choice B confuses the coordinates, swapping x and y values in both intercepts. Choice D has the correct x-intercept but incorrectly shows the y-intercept as $(0, -4)$, which ignores the positive 4 in the original equation. Remember: y-intercepts always have x-coordinate 0, and x-intercepts always have y-coordinate 0. For linear functions in slope-intercept form $y = mx + b$, the y-intercept is simply $(0, b)$.
5. The points $(-1, k)$, $(2, 5)$, and $(5, 11)$ all lie on the same linear function. What is the value of $k$?
   Answer: $-1$ When three points lie on the same linear function, they must have a constant rate of change (slope) between any two points. This means you can use any two points to find the slope, then use that slope with a third point to find unknown values. First, find the slope using the two complete points $(2, 5)$ and $(5, 11)$. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$, so: $m = \frac{11 - 5}{5 - 2} = \frac{6}{3} = 2$ Now use this slope with the points $(-1, k)$ and $(2, 5)$. Since the slope must be the same: $2 = \frac{5 - k}{2 - (-1)} = \frac{5 - k}{3}$ Solving for $k$: $2 \times 3 = 5 - k$, so $6 = 5 - k$, which gives us $k = -1$. Looking at the wrong answers: Choice A ($k = 1$) would give a slope of $\frac{5-1}{3} = \frac{4}{3}$, not 2. Choice C ($k = -4$) would give a slope of $\frac{5-(-4)}{3} = 3$, which is too steep. Choice D ($k = 4$) would give a slope of $\frac{5-4}{3} = \frac{1}{3}$, which is too shallow. The answer is B: $k = -1$. Study tip: When points lie on the same line, always calculate the slope using the two complete points first, then use that slope to find missing coordinates. This two-step approach prevents calculation errors and gives you a clear path to the solution.
6. A linear function is graphed such that it has an x-intercept of $(-4, 0)$ and a y-intercept of $(0, 6)$. What is the value of the function when $x = 2$?
   Answer: $9$ When you see a question about finding the value of a linear function given its intercepts, you're working with the fundamental relationship between points on a line and its equation. First, you need to find the equation of the line using the two given intercepts: $(-4, 0)$ and $(0, 6)$. The slope is $m = \frac{6 - 0}{0 - (-4)} = \frac{6}{4} = \frac{3}{2}$. Since the y-intercept is $(0, 6)$, your equation is $y = \frac{3}{2}x + 6$. Now substitute $x = 2$: $y = \frac{3}{2}(2) + 6 = 3 + 6 = 9$. The correct answer is B. Looking at the wrong answers: Choice A ($3$) represents what you'd get if you forgot to add the y-intercept—just calculating $\frac{3}{2} \times 2 = 3$. Choice C ($12$) is what happens if you accidentally use the wrong slope, perhaps confusing $\frac{6}{4}$ as $3$ instead of $\frac{3}{2}$, giving you $3(2) + 6 = 12$. Choice D ($-3$) results from a sign error, possibly using $-\frac{3}{2}$ as the slope instead of $+\frac{3}{2}$. Study tip: Always double-check your slope calculation by remembering "rise over run" and being careful with positive and negative values. When finding function values, substitute carefully and don't rush through the arithmetic—many errors happen in the final calculation step, not in finding the equation itself.
7. The graph of a linear function passes through points $(-2, 7)$ and $(3, -8)$. What is the rate of change of this function?
   Answer: $-3$ When you see a question asking for the "rate of change" of a linear function, you're being asked to find the slope. The rate of change tells you how much the y-value changes for every unit increase in x. To find the slope between two points, use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Let's use $(-2, 7)$ as our first point and $(3, -8)$ as our second point. Substituting into the formula: $m = \frac{-8 - 7}{3 - (-2)} = \frac{-15}{5} = -3$ The rate of change is $-3$, which matches answer choice D. Let's examine why the other answers are incorrect. Choice A gives $3$, which would result from forgetting the negative sign in your calculation—a common arithmetic error when dealing with negative numbers. Choice B, $-\frac{1}{3}$, comes from accidentally taking the reciprocal of the correct answer and keeping it negative. Choice C, $\frac{1}{3}$, represents taking the reciprocal of the correct slope—this happens when students flip the slope formula and calculate $\frac{x_2 - x_1}{y_2 - y_1}$ instead of $\frac{y_2 - y_1}{x_2 - x_1}$. Remember: slope is always "rise over run" or "change in y over change in x." When working with negative coordinates, be extra careful with your arithmetic, and always double-check that you're subtracting in the same order for both numerator and denominator.

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