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

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!