We have the quadratic function $f(x) = 8x^2 + 16x - 7$. To graph this function and determine the vertex, intercepts, and axis of symmetry, let's break the process down step by step.

Step 1: Find the Vertex

The vertex of a quadratic function in standard form $f(x) = ax^2 + bx + c$ can be found using the vertex formula:

$x_{\text{vertex}} = \frac{-b}{2a}$

For the equation $f(x) = 8x^2 + 16x - 7$:

• $a = 8$
• $b = 16$

Using the formula:

$x_{\text{vertex}} = \frac{-16}{2(8)} = \frac{-16}{16} = -1$

Now substitute $x = -1$ into the function to find the $y$-coordinate of the vertex:

$f(-1) = 8(-1)^2 + 16(-1) - 7 = 8(1) - 16 - 7 = 8 - 16 - 7 = -15$

So, the vertex is $(-1, -15)$.

Step 2: Find the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. For a quadratic function, the axis of symmetry is:

$x = x_{\text{vertex}}$

Thus, the axis of symmetry is:

$x = -1$

Step 3: Find the y-Intercept

To find the $y$-intercept, set $x = 0$ in the equation:

$f(0) = 8(0)^2 + 16(0) - 7 = -7$

Thus, the $y$-intercept is $(0, -7)$.

Step 4: Find the x-Intercept(s)

To find the $x$-intercepts, set $f(x) = 0$ and solve for $x$:

$8x^2 + 16x - 7 = 0$

We can solve this using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting the values $a = 8$, $b = 16$, and $c = -7$:

$x = \frac{-16 \pm \sqrt{16^2 - 4(8)(-7)}}{2(8)}$

$x = \frac{-16 \pm \sqrt{256 + 224}}{16} = \frac{-16 \pm \sqrt{480}}{16}$

$x = \frac{-16 \pm 21.91}{16}$

Thus, the two solutions are:

$x = \frac{-16 + 21.91}{16} = 0.369 \quad \text{(approx.)}$

$x = \frac{-16 - 21.91}{16} = -2.37 \quad \text{(approx.)}$

So, the $x$-intercepts are approximately $(0.369, 0)$ and $(-2.37, 0)$.

Step 5: Domain and Range

• Domain: The domain of any quadratic function is all real numbers, so the domain is:

$(-\infty, \infty)$

• Range: Since the parabola opens upwards (because $a = 8$ is positive), the range starts from the $y$-coordinate of the vertex and goes to infinity. Thus, the range is:

$[-15, \infty)$

Summary:

• Vertex: $(-1, -15)$
• Axis of symmetry: $x = -1$
• $y$-intercept: $(0, -7)$
• $x$-intercepts: $(0.369, 0)$, $(-2.37, 0)$
• Domain: $(-\infty, \infty)$
• Range: $[-15, \infty)$

Would you like to explore the graph further or have any additional questions?

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Here are 5 related questions to expand your understanding:

1. How does the value of $a$ affect the direction in which a parabola opens?
2. What is the significance of the discriminant in the quadratic formula for finding $x$-intercepts?
3. How can we convert a quadratic equation into vertex form?
4. How do changes in the $c$-value affect the graph of a quadratic function?
5. What are the real-world applications of quadratic functions?

Tip: Always double-check the axis of symmetry and vertex when graphing parabolas—they are key to accuracy!