Question Find $x$ -intercepts of the quadratic function $y = x^2 + 8x + 15$ by graphing.

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic function $y = x^2 + 8x + 15$ .
2. To graph the function, we need to find the vertex, axis of symmetry, and the $x$ -intercepts.
3. The $x$ -intercepts are the points where the graph crosses the $x$ -axis, which means the $y$ -value is zero at these points.

STEP 2

The quadratic function is in the standard form $y = ax^2 + bx + c$ . To find the $x$ -intercepts, we need to solve the equation $x^2 + 8x + 15 = 0$ .

STEP 3

We can solve the quadratic equation by factoring, completing the square, or using the quadratic formula. We will attempt to factor first.

STEP 4

To factor the quadratic expression, we look for two numbers that multiply to $c$ (which is 15) and add up to $b$ (which is 8).

STEP 5

The numbers that satisfy this are 3 and 5 because $3 \times 5 = 15$ and $3 + 5 = 8$ .

STEP 6

We can now write the quadratic expression as a product of two binomials using these numbers.
$x^2 + 8x + 15 = (x + 3)(x + 5)$

STEP 7

To find the $x$ -intercepts, we set $y$ to zero and solve for $x$ .
$0 = (x + 3)(x + 5)$

STEP 8

We have two factors equal to zero, which gives us two equations to solve for $x$ .
$x + 3 = 0 \quad \text{and} \quad x + 5 = 0$

STEP 9

Solve the first equation for $x$ .
$x + 3 = 0 \Rightarrow x = -3$

STEP 10

Solve the second equation for $x$ .
$x + 5 = 0 \Rightarrow x = -5$

STEP 11

The $x$ -intercepts of the graph of the quadratic function are $(-3, 0)$ and $(-5, 0)$ .

STEP 12

To graph the quadratic function, we can plot the $x$ -intercepts and the vertex. The vertex can be found using the formula $x = -\frac{b}{2a}$ .

STEP 13

Calculate the $x$ -coordinate of the vertex.
$x = -\frac{b}{2a} = -\frac{8}{2 \cdot 1} = -\frac{8}{2} = -4$

STEP 14

Substitute $x = -4$ into the original quadratic function to find the $y$ -coordinate of the vertex.
$y = (-4)^2 + 8(-4) + 15$

STEP 15

Calculate the $y$ -coordinate of the vertex.
$y = 16 - 32 + 15 = -16 + 15 = -1$

STEP 16

The vertex of the quadratic function is $(-4, -1)$ .

STEP 17

Now we can sketch the graph using the vertex and the $x$ -intercepts. The graph is a parabola opening upwards because the coefficient of $x^2$ is positive.

STEP 18

Draw the axis of symmetry, which is the vertical line passing through the vertex, with the equation $x = -4$ .

STEP 19

Plot the vertex at $(-4, -1)$ and the $x$ -intercepts at $(-3, 0)$ and $(-5, 0)$ on the graph.

STEP 20

Sketch the parabola opening upwards, making sure it passes through the vertex and the $x$ -intercepts.
The $x$ -intercepts of the quadratic function $y = x^2 + 8x + 15$ are $(-3, 0)$ and $(-5, 0)$ . The vertex is $(-4, -1)$ , and the axis of symmetry is the line $x = -4$ .

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