Question

graph the function.
y = 5x² - 20x + 15

use the graphing tool to graph the function
click to enlarge graph

Explanation:

Step1: Analyze the function type

The function \( y = 5x^2-20x + 15 \) is a quadratic function (since the highest power of \( x \) is 2). The general form of a quadratic function is \( y=ax^2+bx + c \), here \( a = 5 \), \( b=- 20 \), \( c = 15 \).

Step2: Find the vertex

The x - coordinate of the vertex of a quadratic function \( y=ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \). Substituting \( a = 5 \) and \( b=-20 \), we get \( x=-\frac{-20}{2\times5}=\frac{20}{10} = 2 \).
To find the y - coordinate, substitute \( x = 2 \) into the function: \( y=5\times(2)^2-20\times(2)+15=5\times4-40 + 15=20-40 + 15=-5 \). So the vertex is at \( (2,-5) \).

Step3: Find the y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function: \( y=5\times(0)^2-20\times(0)+15=15 \). So the y - intercept is at \( (0,15) \).

Step4: Find the x - intercepts (roots)

Set \( y = 0 \), so \( 5x^2-20x + 15=0 \). Divide the entire equation by 5: \( x^2-4x + 3=0 \). Factor the quadratic: \( (x - 1)(x - 3)=0 \). So the roots are \( x=1 \) and \( x = 3 \), which means the x - intercepts are at \( (1,0) \) and \( (3,0) \).

Step5: Determine the direction of the parabola

Since \( a=5>0 \), the parabola opens upwards.
Now, using these key points (vertex \( (2,-5) \), y - intercept \( (0,15) \), x - intercepts \( (1,0) \) and \( (3,0) \)) and the fact that it's a parabola opening upwards, we can graph the function. When using the graphing tool, we can plot these points and draw a smooth curve through them.

(Note: Since the problem asks to use the graphing tool, the actual graphing would involve plotting the vertex, intercepts and drawing the parabola. The key features calculated above help in identifying the shape and position of the graph.)

Answer:

To graph \( y = 5x^2-20x + 15 \), plot the vertex \( (2,-5) \), y - intercept \( (0,15) \), x - intercepts \( (1,0) \) and \( (3,0) \), and draw a parabola opening upwards through these points using the graphing tool.