Graph the equation. $$ y=4 x-3 $$

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Certainly! Below is the graph of the linear equation $$ y = 4x - 3 $$. ### Graph of $$ y = 4x - 3 $$ $$ \begin{tikzpicture}[scale=1] % Draw the axes \draw[<->] (-2,0) -- (4,0) node[right] {$$ x $$}; \draw[<->] (0,-5) -- (0,15) node[above] {$$ y $$}; % Plot the line y = 4x - 3 \draw[domain=-2:4, smooth, variable=\x, blue, thick] plot ({\x}, {4*\x - 3}) node[right] {$$ y = 4x - 3 $$}; % Mark and label points \foreach \x/\xtext in {-2/-2, -1/-1, 0/0, 1/1, 2/2, 3/3, 4/4} \draw (\x,1pt) -- (\x,-3pt) node[below] {$$\xtext$$}; \foreach \y in {-5,0,5,10,15} \draw (1pt,\y) -- (-3pt,\y) node[left] {$$\y$$}; % Highlight y-intercept \fill (0,-3) circle (2pt) node[below left] {$$ (0, -3) $$}; % Highlight another point, say x=1 \fill (1,1) circle (2pt) node[above right] {$$ (1, 1) $$}; \end{tikzpicture} $$ ### Explanation - **Slope and Intercept**: The equation $$ y = 4x - 3 $$ is in the slope-intercept form $$ y = mx + b $$, where: - **Slope ($$ m $$)**: 4. This means for every unit increase in $$ x $$, $$ y $$ increases by 4 units. - **Y-intercept ($$ b $$)**: -3. The graph crosses the y-axis at the point $$ (0, -3) $$. - **Key Points**: - **Y-intercept**: $$ (0, -3) $$ - **Another Point**: By plugging in $$ x = 1 $$, $$ y = 4(1) - 3 = 1 $$, so $$ (1, 1) $$ is another point on the line. - **Graph Behavior**: - The line is straight, extending infinitely in both directions. - As $$ x $$ increases, $$ y $$ increases rapidly due to the steep slope of 4. - As $$ x $$ decreases, $$ y $$ decreases quickly. ### Plotting Points Manually If you'd like to plot the graph manually, follow these steps: 1. **Plot the Y-intercept**: Start by plotting the point $$ (0, -3) $$ on the y-axis. 2. **Use the Slope**: From $$ (0, -3) $$, use the slope to find another point. Since the slope is 4 (which can be written as $$ \frac{4}{1} $$), move up 4 units and right 1 unit to reach $$ (1, 1) $$. 3. **Draw the Line**: Connect these points with a straight line, extending it in both directions. 4. **Label the Axes**: Ensure that both the x-axis and y-axis are labeled and scaled appropriately. ### Visual Representation If you have access to graphing tools like Desmos, GeoGebra, or even graph paper, you can plot the points mentioned above to visualize the line accurately. --- Feel free to ask if you need further assistance with graphing or understanding linear equations! The graph of $$ y = 4x - 3 $$ is a straight line with a slope of 4 and a y-intercept at (0, -3). It passes through points like (0, -3) and (1, 1).

Graph the equation. \[ y=4 x-3 \]

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