Question

graph the function
y = 5x² - 20x + 15
use the graphing tool to graph the function
click to enlarge graph
click the graph, choose a tool in the palette and follow the instr
(graph with grid, axes, and buttons: cancel, save)

Explanation:

Step1: Rewrite the function in vertex form

To graph the quadratic function \( y = 5x^2 - 20x + 15 \), we first complete the square. Factor out the coefficient of \( x^2 \) from the first two terms:
\( y = 5(x^2 - 4x) + 15 \)
Now, complete the square inside the parentheses. Take half of -4, which is -2, square it to get 4. Add and subtract 4 inside the parentheses:
\( y = 5(x^2 - 4x + 4 - 4) + 15 \)
Rewrite the expression as:
\( y = 5((x - 2)^2 - 4) + 15 \)
Distribute the 5:
\( y = 5(x - 2)^2 - 20 + 15 \)
Simplify:
\( y = 5(x - 2)^2 - 5 \)

Step2: Identify the vertex and other key points

From the vertex form \( y = a(x - h)^2 + k \), the vertex is at \( (h, k) \). Here, \( h = 2 \) and \( k = -5 \), so the vertex is \( (2, -5) \).

The coefficient \( a = 5 \) is positive, so the parabola opens upward.

To find the y-intercept, set \( x = 0 \):
\( y = 5(0)^2 - 20(0) + 15 = 15 \), so the y-intercept is \( (0, 15) \).

To find the x-intercepts, set \( y = 0 \):
\( 5x^2 - 20x + 15 = 0 \)
Divide both sides by 5:
\( x^2 - 4x + 3 = 0 \)
Factor:
\( (x - 1)(x - 3) = 0 \)
So the x-intercepts are \( x = 1 \) and \( x = 3 \), which correspond to the points \( (1, 0) \) and \( (3, 0) \).

Step3: Graph the function

Plot the vertex \( (2, -5) \), the y-intercept \( (0, 15) \), and the x-intercepts \( (1, 0) \) and \( (3, 0) \). Then, draw a parabola opening upward through these points.

Answer:

The graph of \( y = 5x^2 - 20x + 15 \) is a parabola with vertex at \( (2, -5) \), opening upward, passing through \( (0, 15) \), \( (1, 0) \), and \( (3, 0) \). (Note: Since this is a graphing problem, the final answer is the graph constructed using the above key points and the shape of the parabola as described.)