To identify the [tex]\( y \)[/tex]-intercept of the quadratic equation [tex]\( y = x^2 - 7x + 10 \)[/tex], follow these steps:
1. Definition of [tex]\( y \)[/tex]-intercept: The [tex]\( y \)[/tex]-intercept of an equation is the point where the graph crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
2. Substitute [tex]\( x = 0 \)[/tex] into the equation: To find the [tex]\( y \)[/tex]-intercept, substitute [tex]\( x = 0 \)[/tex] into the given equation.
[tex]\[
y = (0)^2 - 7(0) + 10
\][/tex]
3. Calculate the result:
[tex]\[
y = 0 - 0 + 10 = 10
\][/tex]
4. Identify the [tex]\( y \)[/tex]-intercept: Therefore, the [tex]\( y \)[/tex]-intercept of the equation [tex]\( y = x^2 - 7x + 10 \)[/tex] is [tex]\( 10 \)[/tex]. This means the graph crosses the [tex]\( y \)[/tex]-axis at the point [tex]\( (0, 10) \)[/tex].
Next, let's graph the equation and match it with the correct answer choice. Here are the steps to graph a quadratic function:
1. Standard Form: The equation [tex]\( y = x^2 - 7x + 10 \)[/tex] is already in standard form [tex]\( y = ax^2 + bx + c \)[/tex].
2. [tex]\( y \)[/tex]-intercept: We already calculated the [tex]\( y \)[/tex]-intercept to be [tex]\( 10 \)[/tex], so the graph will cross the [tex]\( y \)[/tex]-axis at [tex]\( (0, 10) \)[/tex].
3. Find the vertex: The vertex of a quadratic equation in standard form can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex]. For the equation [tex]\( y = x^2 - 7x + 10 \)[/tex]:
[tex]\[
a = 1, \quad b = -7
\][/tex]
[tex]\[
x = -\frac{-7}{2 \times 1} = \frac{7}{2} = 3.5
\][/tex]
4. Substitute the [tex]\( x \)[/tex]-value of the vertex back into the equation to find the [tex]\( y \)[/tex]-coordinate of the vertex:
[tex]\[
y = (3.5)^2 - 7(3.5) + 10
\][/tex]
[tex]\[
y = 12.25 - 24.5 + 10 = -2.25
\][/tex]
So, the vertex is [tex]\( (3.5, -2.25) \)[/tex].
5. Plot more points: Calculate additional points by substituting other values for [tex]\( x \)[/tex] to get corresponding [tex]\( y \)[/tex]-values.
6. Graph the quadratic curve: Using the vertex and the other calculated points, draw the parabola opening upwards since [tex]\( a > 0 \)[/tex].
After plotting, you should verify that the [tex]\( y \)[/tex]-intercept is indeed [tex]\( 10 \)[/tex].
Answer Choice: After graphing the equation, you would match your graph with the given answer choices. The correct answer choice should reflect that the [tex]\( y \)[/tex]-intercept is [tex]\( 10 \)[/tex].
