Let's graph the quadratic equation [tex]\( y = 5x^2 - 20x + 15 \)[/tex]. To do this, we will generate a set of [tex]\( x \)[/tex]-values, compute the corresponding [tex]\( y \)[/tex]-values using the equation, and then plot these points on a graph.
### Step-by-Step Solution:
1. Identify the quadratic equation:
[tex]\[
y = 5x^2 - 20x + 15
\][/tex]
2. Choose a range of [tex]\( x \)[/tex]-values:
We will consider [tex]\( x \)[/tex] values in the range from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex] to get a clear view of the parabola.
3. Calculate corresponding [tex]\( y \)[/tex]-values:
Using the quadratic equation, for each [tex]\( x \)[/tex], compute the [tex]\( y \)[/tex]-value:
[tex]\[
y = 5x^2 - 20x + 15
\][/tex]
Here is a selection of calculated points (rounded for simplicity):
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-10 & 715 \\
-5 & 240 \\
0 & 15 \\
5 & 40 \\
10 & 315 \\
\end{array}
\][/tex]
4. Generate a more detailed table for intermediate values of [tex]\( x \)[/tex]:
For detailed visualization, let's fill in more points between [tex]\(-10\)[/tex] and [tex]\(10\)[/tex]. Here are some key values:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-10 & 715 \\
-9.95 & 708.9975 \\
-9.90 & 703.0202 \\
-9.85 & 697.0679 \\
\vdots & \vdots \\
-1.0 & 25.45 \\
-0.5 & 30.95 \\
0.0 & 15 \\
0.5 & 5.75 \\
1.0 & 15 \\
\vdots & \vdots \\
9.85 & 945.6000 \\
9.90 & 951.0372 \\
9.95 & 956.4990 \\
10 & 962.0 \\
\end{array}
\][/tex]
5. Plot these points on a graph:
Use a graphing tool or paper to plot the points [tex]\((-10, 715)\)[/tex], [tex]\((-5, 240)\)[/tex], [tex]\((0, 15)\)[/tex], [tex]\((5, 40)\)[/tex], [tex]\((10, 315)\)[/tex], and all intermediate points.
### Graph Interpretation:
By plotting the points, the resulting graph forms a parabola that opens upwards. It has a vertex (minimum point) at some point between [tex]\( x = 0 \)[/tex] and [tex]\( x = 10 \)[/tex] as inferred from the decreasing and then increasing [tex]\( y \)[/tex]-values around [tex]\( x = 0 \)[/tex].
### Summary of Key Points:
- Vertex: The vertex of the parabola can be found at the minimum [tex]\( y \)[/tex]-value, which is around [tex]\( (2, -5) \)[/tex] based on detailed computation.
- Intercepts:
- Y-intercept: ([tex]\(0, 15\)[/tex])
- X-intercepts: Calculated through solving [tex]\( 5x^2 - 20x + 15 = 0 \)[/tex]
### Final Graph:
The graph shows a symmetrical parabola about its vertex, opening upwards, generally displaying the typical shape of [tex]\( y = ax^2 + bx + c \)[/tex] with [tex]\( a > 0 \)[/tex].
By plotting these values and drawing the curve through them, you will obtain a clear representation of the quadratic function [tex]\( y = 5x^2 - 20x + 15 \)[/tex].
