Graphing a Function Step by Step

A food delivery driver earns a base wage of $80$ pesos per hour worked. Plot the linear function $f(x) = 80x$, where $x$ is the number of hours and $f(x)$ is the money earned. To draw it, follow these steps:

1. Create a table of values by calculating the outputs for $x = 0, 1, 2$ and $3$.

2. Plot the points on the Cartesian plane.

3. Connect the points with a ray (keeping in mind that hours cannot be negative).

A streaming platform offers a flat rate of $240$ pesos per month, regardless of how many movies you watch. Plot the function $f(x) = 240$, where $x$ is the number of movies.

1. Explain what type of function it is (linear, quadratic, or constant).

2. Determine the value of the function for $x = 0, x = 10$ and $x = 20$.

3. Draw the graph on a set of coordinate axes.

An electric scooter rental company charges $1$ peso to unlock the scooter and $0.50$ pesos for each minute of use. Graph the function $f(x) = 0.5x + 1$, where $x$ is the number of minutes and $f(x)$ is the cost in pesos.

1. a) Calculate the cost for $0$, $5$, $10$ and $20$ minutes.

1. b) Identify the y-intercept and explain what it means in this real-life context.

1. c) Draw the resulting graph. You can use these axes with their grid as a guide (each square is $5$ minutes horizontally and $5$ pesos vertically).

You can do it on paper and send a photo with your answer.

Graph the quadratic function $f(x) = x^2 - 4$. Follow these steps in order:

1. Identify the points where the graph crosses the $X$-axis by solving the equation $x^2 - 4 = 0$.

2. Find the point where the graph crosses the $Y$-axis by calculating $f(0)$.

3. Determine the coordinates of the vertex $V(x_v, y_v)$.

4. Draw the parabola by joining the points above. You can use the axes in the figure, where each division equals 1 unit on both axes.

You can do it on paper and send a photo with your answer.

Draw the parabola $f(x) = -x^2 + 1$. Follow these steps to avoid mistakes:

1. Observe the sign of the coefficient of $x^2$ and decide whether the branches open upwards or downwards.

2. Calculate the vertex. Since there is no term in $x$, what is its x-coordinate $x_v$?

3. Find the intercepts with the $X$ axis.

4. Draw the curve symmetrically with respect to the $Y$ axis.

You can do it on paper and send a photo with your answer.

A basketball player shoots a free throw whose trajectory follows the function $h(t) = -t^2 + 4t + 2$, where $h$ is the height in meters and $t$ is the time in seconds from release. Plot the trajectory of the ball by following these steps:

1. Identify the type of function and determine the direction in which its branches open.

2. Calculate the coordinates of the vertex of the parabola.

3. Find the intercepts with the x-axis ($OX$) and with the y-axis ($OY$).

4. Establish the physical domain of the function taking into account the real-world context (time cannot be negative and the ball stops when touching the ground).

5. Draw the graph using the above data.

Draw the piecewise function step by step:

$f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ 4 & \text{if } x \ge 1 \end{cases}$

1. Calculate the values of the first section for $x = -1, x = 0$ and the value it approaches as $x = 1$.

2. Draw the second section knowing that it is a constant function for values greater than or equal to $1$.

3. Pay attention to the joining point at $x = 1$ to see if there is a jump or if the function is continuous.

You can do it on paper and send a photo with your answer.

A courier company charges its customers based on the weight of packages. The rate is defined by the following piecewise function:

$f(x) = \begin{cases} 2x + 5 & \text{if } 0 \le x < 5 \\ 15 & \text{if } 5 \le x \le 10 \\ x + 5 & \text{if } x > 10 \end{cases}$

where $x$ is the weight in kg and $f(x)$ is the price in pesos. Graph the function step by step:

1. Identify the type of function in each of the three intervals.

2. Calculate the values of the function at the boundary points ($x=5$ and $x=10$).

3. Determine whether the function is continuous at these boundary points by calculating the one-sided limits.

4. Draw each branch of the function over its corresponding interval.

A financial analyst is studying the evolution of the return on a technology investment over 10 years. The function that describes the return $R(t)$ as a function of time $t$ (in years) is given by the following piecewise expression:

$R(t) = \begin{cases} \dfrac{2t - 4}{t + 1} & \text{if } 0 \le t < 4 \\[2mm] |t^2 - 10t + 21| & \text{if } 4 \le t \le 10 \end{cases}$

Draw the graph of the function following these steps:

1. a) Determine the domain of definition in the context of the problem and study whether the function is continuous at $t = 4$ (calculate the left-hand limit and the value $R(4)$).

1. b) Find the points where it crosses the coordinate axes.

1. c) Calculate the asymptotes of the rational branch and decide which ones affect the graph in the interval $0 \le t < 4$.

1. d) Study the absolute value in the second piece: determine for which values of $t$ the expression $t^2 - 10t + 21$ equals zero, write $R(t)$ without absolute value on $[4, 7]$ and on $[7, 10]$, and find the point where the curve touches the $X$ axis forming a peak, as well as the local maximum on $[4, 7]$.

1. e) Graph the function accurately, indicating with an open or closed dot whatever is appropriate at $t = 4$.

You can do it on paper and send a photo with your answer.

You are going to graph the absolute value function $f(x) = |x - 3|$ (in your notebook, if you like, as a support). Follow these steps:

1. First graph the auxiliary line $y = x - 3$ as a dashed line.

2. Identify the point where the line crosses the $X$ axis (it is the point where the direction of the graph changes, the vertex).

3. "Reflect" the part of the line that lies below the $X$ axis onto the positive side of the $Y$ axis.

4. Highlight the final stroke of the function $f(x)$.

Now answer based on your graph:

1. a) What is the abscissa $x$ of the point where the line $y = x - 3$ crosses the $X$ axis?

1. b) What value does the auxiliary line take at $x = 0$? And what value does $f(0)$ take after reflecting?

1. c) Calculate $f(5)$.

1. d) For $x < 3$, the graph of $f$ coincides with the line $y = -x + 3$. What is its slope?

It will be assessed that the auxiliary line, the crossing point, the reflection and the final stroke are consistent with each other, and that the values you give in the parts match your graph.

We want to analyze the evolution of the balance of a youth account that has had sudden movements. The function that describes the estimated balance is $f(x) = |x^2 - 4x|$, where $x$ represents the months elapsed. Draw the graph following this procedure:

1. First, plot the auxiliary parabolic function $g(x) = x^2 - 4x$, finding its vertex and its intersections with the $OX$ axis.

2. Identify the intervals where the value of $g(x)$ is negative.

3. Apply the definition of absolute value to "reflect" the negative part of the graph over the $OX$ axis.

4. Draw the final graph of $f(x)$, making sure there are no values below the x-axis.

You can do it on paper and send a photo with your answer.

Graph the inverse proportionality function $f(x) = \dfrac{4}{x}$.

1. a) Explain why the value $x = 0$ does not belong to the domain.

1. b) Calculate the images for $x = 1, 2, 4$ and their opposites $x = -1, -2, -4$.

1. c) Plot the points obtained on a coordinate system and draw the two branches of the hyperbola, taking into account that the axes are asymptotes.

You can do it on paper and send a photo with your answer.

In the sneaker market, the demand for a specific model is governed by the rational function $q(p) = \frac{200}{p-10}$, where $q$ is the quantity of pairs demanded and $p$ is the price in pesos ($p > 10$). Draw the demand curve by performing the following analysis:

1. Determine the domain of the function in the economic context.

2. Find the equations of the vertical and horizontal asymptotes.

3. Obtain the axis intercepts, if any.

4. Construct a small table of values for significant prices (e.g. $p=20, p=50, p=110$).

5. Plot the function, clearly indicating the demand trend as the price increases.

The average cost of manufacturing a run of customized T-shirts is given by the function $C(x) = \frac{4x + 100}{x}$, where $x$ is the number of T-shirts produced ($x > 0$) and $C(x)$ is the cost per unit in pesos. Graph this rational function by following these steps:

1. a) Simplify the expression of the function by dividing each term of the numerator by the denominator.

1. b) Identify the vertical asymptote and the horizontal asymptote. Explain the economic meaning of the horizontal asymptote.

1. c) Determine whether the function is increasing or decreasing.

1. d) Calculate the average cost for $x=10$, $x=20$ and $x=50$.

1. e) Draw the average cost curve, marking the asymptotes and the points calculated in the previous part.

You can do it on paper and send a photo with your answer.

Given the rational function $f(x) = \frac{x-2}{x^2-4}$, draw its graph following a comprehensive analysis:

1. Determine the domain. Be careful! Identify the type of discontinuity at each point where the denominator equals zero.

2. Simplify the expression of the function and explain what happens graphically at $x = 2$ (is there an asymptote or a "hole"?).

3. Find the vertical and horizontal asymptotes of the simplified function.

4. Study the sign of the function over the intervals defined by the roots and asymptotes.

5. Draw the graph, representing the removable discontinuity as a small empty circle.

You can do it on paper and send a photo with your answer.

The temperature of a new graphics processor is being monitored during a stress test. The temperature $T$ (in $^\circ C$) after $x$ minutes is modeled by the function:

$T(x) = 25 + \dfrac{60x^2}{x^2 + 9}$

To draw the graph that will represent the technical report, carry out the following steps:

1. a) Find the value of the temperature at the initial moment.

1. b) Calculate the horizontal asymptote and explain its physical meaning (limit saturation temperature).

1. c) Determine whether the function has intersection points with the x-axis and justify the result based on the range of the function.

1. d) Study the symmetry of the function and explain how it helps you draw it for hypothetically negative time values.

1. e) Draw the smoothed curve on your axes (you can use a table of values) clearly indicating the asymptote.

You can do it on paper and send a photo with your answer.

An artisan bakery studies its production costs. The unit cost $C(x)$ (in pesos) depends on the number of daily units produced $x$ and follows the function:

$C(x) = \frac{60x^2 + 240}{x - 1}$

Draw a detailed graph of the function for $x > 1$ by carrying out the following analysis:

1. Identify the domain and explain why it makes no sense to analyze values $x \le 1$.

2. Find the vertical asymptote and study the position of the curve relative to it using one-sided limits.

3. Determine whether an oblique asymptote exists using polynomial division and draw it as a construction line.

4. Calculate the intercepts (if any) and use a small table of strategic values to define the curvature.

We wish to sketch the function $f(x) = \frac{|x^2 - 1|}{x}$. Draw the graph following these guidelines:

1. Rewrite the function as a piecewise function by removing the absolute value.

2. Determine the domain and vertical asymptotes.

3. Find the oblique asymptotes for the corresponding branches.

4. Study the symmetry of the function to simplify sketching.

5. Identify the "change of curvature" points where the absolute value acts and draw the final graph.

You can do it on paper and send a photo with your answer.

An archaeological study uses the function $f(x) = \sqrt{x+4} - 2$ to model the depth of an excavation in meters, where $x$ is the horizontal distance from the center of the site. Graph the function by following these steps:

1. Calculate the domain of the function.

2. Find the starting point of the curve (where the domain begins).

3. Find the points of intersection with the coordinate axes.

4. Determine whether the function is increasing or decreasing in its domain.

5. Sketch the graph reflecting the characteristic curvature of square root functions.

Graph the function $f(x) = \sqrt{|x| - 3}$. To do so, follow these logical steps:

1. Determine the domain of the function by solving the inequality $|x| - 3 \ge 0$.

2. Analyse the symmetry of the function. Is it even or odd? How does this affect the construction of the sketch?

3. Study the increase in the interval $(3, +\infty)$.

4. First plot the right side of the graph and then use symmetry to complete the left side.

5. State the intercepts with the $X$-axis.

You can do it on paper and send a photo with your answer.

The number of users of a new social network grows exponentially according to the function $U(t) = 2 \times 1.5^t$, where $U$ is the number of users (in thousands) and $t$ is the time in months. To graph this growth, follow these steps:

1. a) Identify the initial number of users (for $t=0$).

1. b) Determine the type of growth (is it increasing or decreasing?) by looking at the base of the power.

1. c) State the equation of the horizontal asymptote of the function.

1. d) Calculate the values for $t=1$, $t=2$ and $t=3$.

1. e) On Cartesian axes (time $t$ in months on the horizontal axis and users $U$ in thousands on the vertical axis), plot the points obtained and the asymptote, and join them with a smooth curve that represents the evolution of the community.

You can do it on paper and send a photo with your answer.

Draw the graph of the function $f(x) = 2 + e^{1/(x-2)}$ step by step. These types of functions are common in growth studies with critical barriers.

1. Find the domain and the vertical asymptote.

2. Calculate the one-sided limits as $x \to 2^-$ and $x \to 2^+$. What value does the function tend to from the left of the asymptote? And from the right?

3. Determine the horizontal asymptote by calculating the limit as $x \to \pm\infty$.

4. Check whether the function intersects the $X$-axis or the $Y$-axis.

5. Use all the above information to sketch the graph, paying special attention to the discontinuity at $x = 2$.

You can do it on paper and send a photo with your answer.

The sound intensity (on a simplified logarithmic scale) perceived by a microphone is given by $S(x) = \log_2(x) + 3$, where $x$ is the power of the sound source. Graph the function $S(x)$ by doing the following:

1. a) Establish the domain of the function.

1. b) Identify the vertical asymptote.

1. c) Calculate the point where the intensity is zero (point of intersection with the $OX$ axis).

1. d) Obtain the value of the function for key powers such as $x=1$, $x=2$ and $x=4$.

1. e) Draw the graph showing the logarithmic behavior, indicating on it the asymptote, the point of intersection with $OX$ and the points from part d).

You can do it on paper and send a photo with your answer.

The net profit $B(x)$ of a mobile app startup (in thousands of pesos) follows the model:

$B(x) = \begin{cases} x^2 + 4x & \text{if } x < 0 \\ 5 \times \ln(x + 1) & \text{if } x \ge 0 \end{cases}$

Where $x$ represents the advertising investment balance (negative is debt, positive is net investment). Draw the graph step by step:

1. Study the continuity at the critical point $x = 0$.

2. For the left-hand piece ($x < 0$), find the vertex of the parabola and its points of intersection with the axes.

3. For the right-hand piece ($x \ge 0$), study the growth and find the point where it crosses the $Y$ axis. Also calculate at least one more point (for example, for $x = 4$) to help you draw the curve.

4. Determine whether the function has any asymptotes.

5. Draw both branches on the same axes and mark the joining point.

You can do it on paper and send a photo with your answer.