Answer :
To sketch the graph of the quadratic function [tex]\( y = x^2 - 7x + 12 \)[/tex], we need to follow a series of steps, including finding the key features of the graph such as the roots (the x-intercepts), the vertex, and the y-intercept. Let's break this down step-by-step:
1. Determine the roots (x-intercepts) of the function:
To find the roots of the quadratic equation, we solve [tex]\( x^2 - 7x + 12 = 0 \)[/tex].
The quadratic can be factored as:
[tex]\[ x^2 - 7x + 12 = (x - 3)(x - 4) \][/tex]
Setting each factor equal to zero gives:
[tex]\[ x - 3 = 0 \quad \text{or} \quad x - 4 = 0 \][/tex]
Thus, the roots are:
[tex]\[ x = 3 \quad \text{and} \quad x = 4 \][/tex]
2. Determine the vertex of the parabola:
The vertex form of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be used to find the vertex. The x-coordinate of the vertex is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For [tex]\( y = x^2 - 7x + 12 \)[/tex], [tex]\( a = 1 \)[/tex] and [tex]\( b = -7 \)[/tex]. Thus,
[tex]\[ x = -\frac{-7}{2 \times 1} = \frac{7}{2} = 3.5 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 3.5 \)[/tex] into the equation:
[tex]\[ y = (3.5)^2 - 7(3.5) + 12 \][/tex]
Calculate each term separately:
[tex]\[ (3.5)^2 = 12.25, \quad -7(3.5) = -24.5, \quad \text{and} \quad y = 12.25 - 24.5 + 12 \][/tex]
[tex]\[ y = 12.25 - 24.5 + 12 = -0.25 \][/tex]
Thus, the vertex is at:
[tex]\[ (3.5, -0.25) \][/tex]
3. Determine the y-intercept:
The y-intercept is where the graph crosses the y-axis, which occurs when [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = 0^2 - 7(0) + 12 = 12 \][/tex]
Therefore, the y-intercept is at:
[tex]\[ (0, 12) \][/tex]
4. Sketch the graph:
- Plot the roots (x-intercepts) at [tex]\( (3, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
- Plot the vertex at [tex]\( (3.5, -0.25) \)[/tex].
- Plot the y-intercept at [tex]\( (0, 12) \)[/tex].
- Draw a smooth curve to represent the parabola, connecting these points.
Since the coefficient of [tex]\( x^2 \)[/tex] is positive ([tex]\( a = 1 \)[/tex]), the parabola opens upwards.
### Final Sketch
Here is the rough sketch of the graph of [tex]\( y = x^2 - 7x + 12 \)[/tex]:
- The vertex is at [tex]\( (3.5, -0.25) \)[/tex].
- The roots are at [tex]\( (3, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
- The y-intercept is at [tex]\( (0, 12) \)[/tex].
[tex]\[ \begin{array}{l} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad y\\ \quad 12\_/\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \\ \quad \quad \quad \quad \quad \quad \quad\_/\quad \cup \quad \quad \quad \quad \quad \quad \quad \quad \quad \\ \quad \quad \quad \quad \,1\_/\quad\quad\_/\quad \\ (0,12)\quad \quad(3,0)\quad(4,0)\\ \quad \quad \quad \quad \quad(3.5, -0.25)\\ \quad \quad \quad \quad\quad \quad \quad\quad \quad \quad\quad \quad\quad\quad\quad\quad\quad\quad x\\ \end{array} \][/tex]
1. Determine the roots (x-intercepts) of the function:
To find the roots of the quadratic equation, we solve [tex]\( x^2 - 7x + 12 = 0 \)[/tex].
The quadratic can be factored as:
[tex]\[ x^2 - 7x + 12 = (x - 3)(x - 4) \][/tex]
Setting each factor equal to zero gives:
[tex]\[ x - 3 = 0 \quad \text{or} \quad x - 4 = 0 \][/tex]
Thus, the roots are:
[tex]\[ x = 3 \quad \text{and} \quad x = 4 \][/tex]
2. Determine the vertex of the parabola:
The vertex form of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be used to find the vertex. The x-coordinate of the vertex is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For [tex]\( y = x^2 - 7x + 12 \)[/tex], [tex]\( a = 1 \)[/tex] and [tex]\( b = -7 \)[/tex]. Thus,
[tex]\[ x = -\frac{-7}{2 \times 1} = \frac{7}{2} = 3.5 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 3.5 \)[/tex] into the equation:
[tex]\[ y = (3.5)^2 - 7(3.5) + 12 \][/tex]
Calculate each term separately:
[tex]\[ (3.5)^2 = 12.25, \quad -7(3.5) = -24.5, \quad \text{and} \quad y = 12.25 - 24.5 + 12 \][/tex]
[tex]\[ y = 12.25 - 24.5 + 12 = -0.25 \][/tex]
Thus, the vertex is at:
[tex]\[ (3.5, -0.25) \][/tex]
3. Determine the y-intercept:
The y-intercept is where the graph crosses the y-axis, which occurs when [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = 0^2 - 7(0) + 12 = 12 \][/tex]
Therefore, the y-intercept is at:
[tex]\[ (0, 12) \][/tex]
4. Sketch the graph:
- Plot the roots (x-intercepts) at [tex]\( (3, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
- Plot the vertex at [tex]\( (3.5, -0.25) \)[/tex].
- Plot the y-intercept at [tex]\( (0, 12) \)[/tex].
- Draw a smooth curve to represent the parabola, connecting these points.
Since the coefficient of [tex]\( x^2 \)[/tex] is positive ([tex]\( a = 1 \)[/tex]), the parabola opens upwards.
### Final Sketch
Here is the rough sketch of the graph of [tex]\( y = x^2 - 7x + 12 \)[/tex]:
- The vertex is at [tex]\( (3.5, -0.25) \)[/tex].
- The roots are at [tex]\( (3, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
- The y-intercept is at [tex]\( (0, 12) \)[/tex].
[tex]\[ \begin{array}{l} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad y\\ \quad 12\_/\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \\ \quad \quad \quad \quad \quad \quad \quad\_/\quad \cup \quad \quad \quad \quad \quad \quad \quad \quad \quad \\ \quad \quad \quad \quad \,1\_/\quad\quad\_/\quad \\ (0,12)\quad \quad(3,0)\quad(4,0)\\ \quad \quad \quad \quad \quad(3.5, -0.25)\\ \quad \quad \quad \quad\quad \quad \quad\quad \quad \quad\quad \quad\quad\quad\quad\quad\quad\quad x\\ \end{array} \][/tex]
